design of immersed tunnel - aalborg universitetprojekter.aau.dk › projekter › files › 14460087...

M o d e l a c c u r a c y i n a s e i s M i c d e s i g n o f i M M e r s e d t u n n e l

Master’s Thesis, 2008 J a k o b H a u s g a a r d L y n g s

Master of Science in Civil and Structural EngineeringT h e S c h o o l o f C i v i l E n g i n e e r i n g , A a l b o r g U n i v e r s i t y

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Model accuracy in aseismic designof immersed tunnel

MASTER’ S THESIS

THE SCHOOL OFCIVIL ENGINEERING

MASTER OFSCIENCE IN CIVIL AND STRUCTURAL ENGINEERING

AALBORG UNIVERSITY, 2008

Jakob Hausgaard Lyngs

PREFACE

This master’s thesis is prepared at the Master of Science Programme in Civil andStructural Engineering at Aalborg University, Denmark. The thesis is the outcome ofa long candidate project on the 3rd and 4th semester. The subject for the project periodis Design and Analysis of Advanced/Special Structures.

The thesis consists of the following parts:

• Main thesis• Appendices. The appendices are located in the back of the main thesis.• DVD with developed programs, data files, animations, and a pdf-version of the

thesis with working hyperlinks. The DVD is found in the back of the thesis.

The thesis project has been supervised by Associate Professor Lars Andersen, whoseinvaluable inspiration and tutoring have been greatly appreciated. Furthermore, thethesis work has been inspired by the real-world immersed tunnel design experienceand knowledge of Senior Project Manager Michael Tonnesen, COWI, and R&D Man-ager Carsten S. Sørensen, COWI. I would like to thank them for their highly appreci-ated guidance.

The thesis may also be found on the websitefinalthesis.hausgaard-lyngs.dk.

Aalborg, 11th June 2008 Jakob Hausgaard Lyngs

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finalthesis.hausgaard-lyngs.dk

ABSTRACT

This thesis deals with the model accuracy for seismic designof immersed tunnels, likethe proposed underwater artery in Thessaloniki, Greece.

The soil response to seismic waves is analysed in the frequency domain by meansof the domain transformation method and the finite element method. The seismic re-sponse of an immersed tunnel and the damage in the gasket joints have been calculatedwith a closed form solution, a Winkler-type model, and a fullthree-dimensional con-tinuum model. The Winkler model and the continuum model are applied in the timedomain.

Focus is especially given to the diverging results from the Winkler model, commonlyused in seismic design, and the continuum model, which is considered to be more ac-curate. Through comparative analyses it is shown that the presented Winkler modelis not able to model retroaction from the tunnel to the soil. This entails that the pre-sented Winkler model is not suited for seismic design of an immersed tunnel withnon-uniform cross section.

Sensitivity analyses are performed to analyse the influenceof the many parameterswhich must be determined for a seismic design. It is shown that the stratification andthe soil parameters, together with the earthquake magnitude, influence significantly onthe tunnel damage. On the other hand, the modelling of the immersed tunnel gasketjoints has very little influence on the calculated gasket deformation. Finally, it is shownthat the critical direction of wave propagation is an angle of approximately45◦ to thetunnel axis.

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RESUMÉ

Dette afgangsprojekt omhandler modelusikkerheder ved jordskælvsdesign af sænke-tunneler, med en sænketunnel i Thessaloniki i Grækenland som case.

Jordens respons som følge af jordskælvsbølger er analyseret i frekvensdomæne vedhjælp af domænetransformationsmetoden og finite element metoden. Responset afsænketunnelen og skaderne i koblingerne mellem tunnelelementerne er beregnet meden løsning på sluttet form, en model af Winkler-typen samt enfuld tredimensionelkontinuummodel. Analyserne med de sidste to modeller foregår i tidsdomæne.

Winkler-modellen anvendes ofte ved jordskælvsdesign, mens kontinuummodellen eranset for at være mere nøjagtig. De to modeller giver meget forskellige resultater, ogårsagen til denne forskel er undersøgt nøje. Gennem sammenlignende analyser er detvist, at den opstillede Winkler-model ikke er i stand til at modellere tilbagekobling fratunnelen til jorden. Dette indebærer at den opstillede Winkler-model ikke er egnet tilat designe en sænketunnel med fleksible koblinger mod jordskælv.

Ved design mod jordskælv er der mange parametre der skal bestemmes. For at under-søge hvor nøjagtigt disse parametre bør bestemmes er der udført følsomhedsanalyser.Det er vist at lagdelingen, jordens parametre samt jordskælvets størrelse har stor ind-flydelse på den beregnede skade på tunnelen. På den anden sidehar det meget lille be-tydning for skaden på tunnelen, hvordan det vælges at modellere koblingerne mellemtunnelelementerne. Endelig er det vist at den kritiske udbredelsesretning for jord-skælvsbølgen er en vinkel på omtrent45◦ med tunnelens længdeakse.

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CONTENTS

Preface i

Abstract iii

Resumé v

1 Introduction 1

1.1 Thessaloniki immersed tunnel . . . . . . . . . . . . . . . . . . . . . .. . 1

1.2 The thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Demarcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Conclusion 5

I Introduction and definitions 9

3 Immersed tunnels 11

3.1 Construction techniques . . . . . . . . . . . . . . . . . . . . . . . . . .. 11

3.1.1 Construction and transport . . . . . . . . . . . . . . . . . . . . . 11

3.1.2 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Advantages of immersed tunnels . . . . . . . . . . . . . . . . . . . . .. 14

4 Earthquakes 17

4.1 Plate tectonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Wave types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Determination of design earthquake . . . . . . . . . . . . . . . . .. . . 21

4.3.1 Size of earthquake . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3.2 Seismic hazard analysis . . . . . . . . . . . . . . . . . . . . . . . 22

4.3.3 Code spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3.4 The present thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 The influence of earthquakes on underground structures .. . . . . . . . 25

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4.5 Damage modes of tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.6 Incoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Design Basis 27

5.1 Geometry of tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Geotechnical parameters . . . . . . . . . . . . . . . . . . . . . . . . . .. 28

5.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3.1 Loss factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3.2 Viscous damping . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3.3 Hysteretic damping . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3.4 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.4 Design cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.5 Input motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.5.1 Time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.5.2 Apparent velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5.3 Direction of propagation . . . . . . . . . . . . . . . . . . . . . . 35

5.6 Stiffness of gaskets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

5.6.1 Longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . 35

5.6.2 Shear stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.7 Damage criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

II Methods of analyses 41

6 Wave propagation through soil 43

6.1 Generalized geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2 The domain transformation method . . . . . . . . . . . . . . . . . . .. 45

6.3 The finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.3.1 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.3.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3.4 Choice of calculation method . . . . . . . . . . . . . . . . . . . 48

6.4 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.5 Earthquake response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.5.1 Frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.5.2 Time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Closed form solution 53

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7.1 Axial strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.2 Deformation at gaskets . . . . . . . . . . . . . . . . . . . . . . . . . . . .54

7.3 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8 Winkler model 57

8.1 Model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.2.1 Soil spring stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.2.2 Gasket stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8.2.3 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.2.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.3 Input motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.4.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

8.4.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

9 Continuum model 67

9.1 Parts and meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

9.1.1 Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9.1.2 Gaskets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9.1.3 Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

9.1.4 Discretization considerations . . . . . . . . . . . . . . . . . .. 69

9.2 Material modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

9.2.1 Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.2 Tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.3 Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

9.2.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

9.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

9.3 Earthquake loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

9.4 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9.4.1 Ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

9.4.2 Master and slave surfaces . . . . . . . . . . . . . . . . . . . . . 75

9.5 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.5.1 Deformation at corners . . . . . . . . . . . . . . . . . . . . . . . 76

9.5.2 Degrees of freedom equivalent to Winkler model . . . . . .. . 76

9.6 Verification of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

9.6.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 78

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9.6.2 Representation of free-field soil deformation . . . . . .. . . . . 81

9.6.3 Time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

III Analyses of methods and parameters 85

10 Comparison of models 87

10.1 Initial results from design basis . . . . . . . . . . . . . . . . . .. . . . . 87

10.1.1 Closed form solution . . . . . . . . . . . . . . . . . . . . . . . . 87

10.1.2 Winkler model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.1.3 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.1.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

10.2 The Winkler model vs. the continuum model . . . . . . . . . . . .. . . 91

10.2.1 Crude errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

10.2.2 Three-dimensional wave propagation . . . . . . . . . . . . .. . 92

10.2.3 Deformation in gasket . . . . . . . . . . . . . . . . . . . . . . . 92

10.2.4 Importance of retroaction . . . . . . . . . . . . . . . . . . . . . .94

10.2.5 Modelling without gaskets . . . . . . . . . . . . . . . . . . . . . 99

10.3 Winkler soil springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100

10.4 Prestress in tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

11 Soil parameters and stratification 103

11.1 Method of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

11.2 Impact of layer depths . . . . . . . . . . . . . . . . . . . . . . . . . . . .104

11.2.1 Thickness of layer A . . . . . . . . . . . . . . . . . . . . . . . . 104

11.2.2 Thickness of layer B . . . . . . . . . . . . . . . . . . . . . . . . 105

11.2.3 Distance to bedrock . . . . . . . . . . . . . . . . . . . . . . . . . 105

11.3 Wave velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

11.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

11.4.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

11.4.2 Hysteretic damping and viscous damping . . . . . . . . . . .. 108

12 The earthquake 111

12.1 Apparent velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111

12.2 Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

12.3 Earthquake amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . .116

13 Modelling of gasket joints 117

13.1 Longitudinal stiffness . . . . . . . . . . . . . . . . . . . . . . . . . .. . 117

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Contents

13.2 Shear stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

Bibliography 123

Appendices 127

A Derivation of the domain transformation method 129

A.1 Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.2 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A.3 Transformation into frequency domain . . . . . . . . . . . . . . .. . . . 130

A.4 Relation between the layers . . . . . . . . . . . . . . . . . . . . . . . .. 131

A.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A.6 Solution for inner interfaces . . . . . . . . . . . . . . . . . . . . . .. . . 133

B Derivation of the finite element method for wave propagation 135

B.1 Geometry and topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.2 Element matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.2.1 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.2.2 Mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.3 Solution of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

C Enhancements to FE Winkler model 139

C.1 Damping and mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . 139

C.2 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

C.3 Forced displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

C.4 Spring elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

D Cross section data 143

D.1 Length of gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

D.2 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

D.3 Second moment of area . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

E Rotation matrices 145

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1I NTRODUCTION

In this chapter, a brief outline of the Thessaloniki immersed tunnel project and thepresent thesis project is given.

1.1 Thessaloniki immersed tunnel

Thessaloniki is the second-largest city in Greece with a population around one millioninhabitants. The location of Thessaloniki is depicted in Figure 1.1. Thessaloniki islocated in the Axios-Vardaris zone, adjacent to the Servomakedonian zone, which ischaracterized as one of the most active seismotectonic zones in Europe. Several activefaults are present in the region. (Pitilakiset al.2007, p134)

Thessaloniki

Figure 1.1: Location of Thessaloniki.

This project deals with the seismic design process of a proposed immersed tunnel,planned as a 6-lane road toll-tunnel. The intended locationis shown in Figure 1.2on the following page. The principal objective of the project is to provide congestionrelief to the centre of Thessaloniki by the creation of an underground by-pass. The

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Chapter 1. Introduction

heavily trafficked avenue on the seafront is to be pedestrianised upon the completionof the tunnel project.

Figure 1.2: The location of the immersed tunnel in the Thermaikos Gulf outside the city cen-tre of Thessaloniki. The tunnel is depicted with a red dashed line. (Google Earth2008)

The immersed tunnel will be about1.2km in length and is placed on the seabed at awater depth of around10m. In both ends, the immersed tunnel is linked to cut & covertunnels. In Figure 1.3 a sketch of the immersed tunnel on the seabed is shown.

Figure 1.3: Illustration of the Thessaloniki tunnel. (COWI 2008)

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The thesis

1.2 The thesis

The focus in the present thesis is the accuracy of the presented calculation models forseismic design of immersed tunnels. While the Thessaloniki immersed tunnel is usedas case for the thesis project, a final design for the specific tunnel is not provided.Effort is concentrated on the development and analysis of five calculation models: thedomain transformation method and the finite element method for wave propagation,and a closed form solution, a Winkler-type model, and a full three-dimensional con-tinuum model for the soil-structure interaction.

The thesis consists of three parts. In Part 1 the basis for thethesis is formed, as thegeneral concepts of immersed tunnels and earthquakes are presented, and a basis ofdesign parameters is found.

In Part 2 the calculation methods are presented. The domain transformation methodand the finite element method have been derived and used for a one-dimensional cal-culation of the wave propagation from bedrock to the level atthe immersed tunnel.Thereafter, the closed form solution is presented, followed by the Winkler model andthe continuum model.

Part 3 encloses analyses and comparisons of the models. The reasons for divergencesin the obtained results are discussed, and sensitivity analyses are performed to clarifythe influence of various parameters on the tunnel damage.

Following the bibliography, appendices containing lengthy or trivial derivations notsuited for the main thesis are gathered.

1.3 Demarcation

To maintain focus on the analysis of the accuracy of the calculation models and toprovide a feasible workload, the demarcation of this thesishas elided many potentialphenomena. Some of these are:

• Gasket compression variationsTemperature variations will cause the length of the tunnel elements to vary,thereby changing the compression of the gasket joints between the elements.Furthermore, due to relaxation, the compression stress in the gaskets will de-crease in time. Both of these effects are disregarded.

• Cut & cover tunnelsThe thesis purely deals with the immersed tunnel. The cut & cover tunnelshave not been regarded, nor have the connections between theimmersed tunnelelements and the cut & cover tunnels been given any considerations.

• S-wavesAs it is discussed in Section 4.2 on page 19, only S-waves are regarded for thecalculations of the wave propagation.

• No permanent deformationIn the analyses, only transient earthquake motion is analysed. Thus any per-manent deformation – which could steme.g.from a fault displacement directly

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Chapter 1. Introduction

beneath the tunnel – is disregarded.

• Damage related to the gasket jointsOnly the deformation in the gaskets are defined as damage in the analyses, asit is discussed in Section 5.7. Thus, the stresses in the tunnel elements arenot given any consideration. This is done sincee.g.Vrettoset al. (2007) andTonnesen (2008) state that the gaskets are normally the critical spots for seismicanalyses of immersed tunnels.

• LinearizationEven though some rather harsh non-linear problems are dealtwith, all of themodels presented in this thesis are purely linear. This is chosen in order to re-duce computation time, and since it is deemed that only smallstrains will occurin the dynamic analyses. Also, to utilise the frequency domain, linear materialbehaviour is a prerequisite. Where very non-linear behaviour is occurring,e.g.for the soil and for the gasket behaviour, sensitivity analyses are performed inPart 3 to quantify the possible error.

Also, the use of linear material models entails that nominaltensile stresses willoccur in the soil. This is not compatible with the general observed behaviour of(non-cohesive) soils. However, the soil pressures due to gravity are not incorpo-rated in the models, since the dynamic analysis only models oscillations aroundthe state of equilibrium. Thus, the real stresses will most likely vary betweenmore or less compression, making the linear material modelsacceptable froman engineering point of view.

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2CONCLUSION

In the following, the conclusions of the thesis are summarized.

Part 1

In Part 1 the common basis for the thesis has been established.

The general concepts of immersed tunnels and earthquakes are presented, and the de-sign cross section has been established on the basis of microzonation by Anastasiadiset al. (2001) and data from design reports by COWI (2007). An acceleration timeseries from the 1995 Aegion earthquake has been used as the seismic input, since aseismic hazard analysis is outside the scope of this thesis.

Hysteretic and viscous damping have been applied in the models, and the differencesbetween the mechanisms are discussed. Furthermore, the behaviour of the gasketjoints under tunnel axial and cross axial loading has been presented, and the lineariza-tion has been discussed. Finally, the damage criterion has been defined as the openingand compression of the gasket joints, which are calculated in the following parts.

Part 2

In Part 2 the applied calculation models are presented.

The wave propagation from bedrock to the level at the tunnel has been calculated inthe frequency domain with the domain transformation methodand the finite elementmethod, both of which are derived. It has been shown that the models produce identi-cal results if the finite element method is discretized sufficiently. But since the domaintransformation method is based on an analytical solution itis more computational ef-ficient and is used in general in the thesis.

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Chapter 2. Conclusion

The gasket deformation has been calculated with three different methods. A closedform solution is presented as a very simple way of estimatingthe order of magnitudeof the gasket deformation, based on the free-field soil response.

Subsequently a Winkler-type model has been established, where the immersed tunnelelements are equated with beams, and the soil-structure interaction is modelled withlinear springs. The Winkler model has been solved in the timedomain by means of afinite element code, which has been developed for this purpose. The model is codedin MATLAB , and the spring stiffness’s are determined withABAQUS andPLAXIS.

Finally, a full three-dimensional continuum model is presented. The model is coded inthe commercial finite element codeABAQUS with use of user subroutines and solvedin the time domain. The postprocessing is done withMATLAB.

Part 3

In Part 3 the analyses of the thesis are gathered.

The three different ways of calculating the gasket deformation have been held upagainst each other. The three models yield very different results. The closed formsolution provides nearly twice the deformation of the Winkler model, which further-more calculates deformations of more than 10 times the deformation of the continuummodel.

It may in particular cause surprise that the Winkler model and the continuum modeldiffer with an order of magnitude. The continuum model is regarded as the moreaccurate model of the physical problem, while the Winkler model is commonly usedfor seismic design of immersed tunnels. Therefore this divergence has been analysed.

It has been shown that no crude errors have been made in the coding of the models,and that the models yield very similar output if the tunnel ismodelled with a uniformcross section. However, the immersed tunnel has a non-uniform cross section, as itconsists of concrete tunnel elements which are connected with rubber gaskets.

Through comparative analyses it has been found that the Winkler model fails to modelany retroaction from the tunnel to the soil, and this specificproperty is shown to bethe single most significant effect on the gasket deformation. In the continuum modelthe gasket deformation increases dramatically if retroaction is obstructed. It has beenbriefly outlined how retroaction could be implemented in theWinkler model, and itis the conclusion that without this enhancement, the Winkler model isnot suited formodelling of structures with non-uniform cross section, such as the immersed tunnel.

The deformation modes of the immersed tunnel have also been analysed, and it hasbeen found that with the chosen damage criterion, the dominating mode is axial com-pression and extension, illustrated in Figure 4.10a on page25. Also, bending of thetunnel provides some gasket deformation, but only approximately10% of the total de-formation.

During design of the immersed tunnel subjected to earthquake strong ground motion,many different parameters must be determined. To analyse the importance of the ac-

6

Part 3

curacy of these parameters, sensitivity analyses have beencarried out. The thickness,wave velocity and damping parameters of the subsoil layers have been analysed, to-gether with the apparent velocity, angle of propagation anddisplacement amplitude ofthe earthquake. Finally, the gasket joint behaviour is analysed.

The results from the sensitivity analyses indicate first andforemost, that the generallinearization is acceptable. The gasket behaviour is in principle very non-linear, butit is shown that the gasket deformation is highly insensitive to variations in the gasketstiffness. Thus, only little effort should be given to the determination of the gasketstiffness for a final design. Furthermore, the general non-linear behaviour of soil willnot influence significantly on the gasket deformation, cf. Section 10.3.

The sensitivity analyses further show, that the most important parameters to define arethose related to the ground conditions at the project site. The stratification and thewave speed may exert significant influence on the gasket deformation. The analysesshow that the eigenfrequency of the soil column influences the results. Thus, it is notpossible to choose the stratification and the soil parameters on the safe side withoutperforming sensitivity analyses. Furthermore, the determination of the correct designearthquake obviously will exert significant influence on thecalculated tunnel damage.

The influence of the apparent velocity – the observed propagation velocity of the earth-quake wave front at the ground surface – is only moderate according to the sensitivityanalysis. For increasing apparent velocity the tunnel damage decreases, and there-fore the apparent velocity for a final design should be chosenas the lowest reasonablevalue. Additionally, it has been shown that the critical direction of wave propagationis oblique to the tunnel with an angle of approximately45◦. Finally, it is shown thatthe chosen damping parameters only have very little influence on the tunnel damage.

It should be noted, that for normalized deformation, the continuum model, the Winklermodel, and, where applicable, the closed form solution, provide very similar resultsfor most of the sensitivity analyses. Thus, even though the Winkler model and theclosed form solution in general provide overly conservative results for the gasket de-formation, these models can be used for some parameter analyses.

7

PARTII NTRODUCTION AND DEFINITIONS

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3I MMERSED TUNNELS

Tunnels are, in general, constructions used to pass under soil, mountains or water.The construction techniques are many and count among othersbored tunnels, cut &cover tunnels, NATM tunnels, blasted tunnels and floating tunnels, each type of tunnelhaving its specific strengths. The choice of tunnel type dependsi.a. on economy, thegeography of the project site, and the construction time. InThessaloniki, an immersedtunnel connected with cut & cover tunnels on shore is under design. In the presentchapter, the immersed tunnel type will be described.

3.1 Construction techniques

An immersed tunnel consists of prefabricated tunnel elements, made so that whentemporary bulkheads are applied in both ends, a tunnel element can be floated to theproject site. There, the tunnel elements are, one by one, lowered into an excavatedtrench in the seabed. Finally, the trench are backfilled around the tunnel.

The construction methods of immersed tunnels are developedin the beginning of the1900s in the USA. The cross sections were at that time typically circular and of steel.In the 1930s rectangular cross sections of reinforced concrete were developed in Eu-rope. (DGF 2005, p137)

3.1.1 Construction and transport

The tunnel elements are constructed on shore, and are typically of 100m to 180m inlength. A suitable area of the right size and connected with the project site with water-ways has to be found. A veritable dry dock is built by damming the construction area,as it can be seen in Figure 3.1 on the next page. With respect toan optimization based

11

Chapter 3. Immersed tunnels

on time and economy, the dry dock should be able to contain all, or some fraction, ofthe tunnel elements.

Figure 3.1: The Conwy tunnel in Wales during construction (1988). The connection tunnelsare fully constructed, and plugged with bulkheads. The completed immersed tun-nel elements in the dry dock can be seen in the lower left corner. (DGF 2005,p140)

When the construction of a series of tunnel elements has finished, temporary bulkheadsare applied to the ends of the tunnel elements. The tunnel elements are now floatedoff to the project site, either on barges or simply, if the tunnel elements have positivebuoyancy, with the help of tugboats, as it can be seen in Figure 3.2.

Figure 3.2: Tug boats manoeuvring an immersed tunnel element. (ITA 1999)

3.1.2 Installation

When arrived at the project site, the tunnel elements are installed one by one. Thetunnel element is ballasted with water until negative buoyancy exists, and the tun-

12

Installation

nel element is lowered into the excavated trench with the help of cranes mounted onbarges. The lowering process is illustrated in Figure 3.3.

Ballast concrete

Figure 3.3: Lowering of a immersed tunnel element. After Trelleborg (2007).

On one of the tunnel ends, a gasket is mounted. The purpose of this gasket is to sealthe connection between two adjacent tunnel elements, ensuring the water tightness ofthe structure. The gaskets used in the tender design in Thessaloniki areGina gas-ket profiles, manufactured by Trelleborg Bakker, The Netherlands. (COWI 2007) InFigure 3.4, a cross section and a mounted gasket are depicted.

Mounted

toelem

ent

Com

pressedtow

ardsnextelem

ent

(a) Cross section of Gina-profile typeETS 180-220-SN

(b) Gina gasket mounted to a tunnel ele-ment. The bulkhead is installed

Figure 3.4: Gina gasket profile. (Trelleborg 2007)

The tunnel element is placed adjacent to the previously installed element (or the cut& cover tunnel in Thessaloniki, if it is the first element to beplaced). The steps in thecoupling of the two tunnel elements are illustrated in Figure 3.5.

The tunnel elements are pulled against each other, and the gasket compresses andforms a reservoir between the bulkheads, cf. Figure 3.5b and3.5c. When the reservoiris emptied and filled with air at atmospheric pressure, the hydrostatic pressure at thefree end of the tunnel element under installation compresses the Gina gasket, as it isshown in Figure 3.5d.

Finally, to ensure that the tunnel element stays on the seabed, ballast concrete is castin the tunnel element, as it is depicted in Figure 3.3.

13

Chapter 3. Immersed tunnels

(a) The immersed tunnel element is pulledagainst the previously installed one

(b) A small reservoir is created (c) Between thebulkheads (ITA1999)

(d) Water is pumped out and the hydro-static pressure compresses the Ginaprofile

(e)The bulkheads are removed, and asecondary water seal is applied

Figure 3.5: Coupling of immersed tunnel with Gina gasket. After Trelleborg (2007).

3.2 Advantages of immersed tunnels

Immersed tunnels should be considered whenever it is neededto cross water. Typical,the choice will stand between a bored tunnel, an immersed tunnel, or a bridge. Someof the advantages of immersed tunnels are:

• AlignmentSince an immersed tunnel is placed on the seabed, its total length will be lessthan the length of a bored tunnel, thus reducing costs.

• Cross sectionSince the immersed tunnel elements are constructed on shore, it is possible toconstruct various cross sections. The cross section of a bored tunnel is normallyrestricted to being circular.

• Ground conditionsWhile both a bridge and a bored tunnel requires relative good ground conditions,it is possible to install an immersed tunnel in most types of soil, including softalluvial materials. However, in relation to the topic for the present thesis itshould be noted that soft soil may amplify seismic waves significantly.

• Land availabilityEven though immersed tunnel construction requires much space for the con-struction dry dock, this could be located relative far from the project site, makingit possible to construct an immersed tunnel in urban areas, such as Thessaloniki.

• Reclamation

14

Advantages of immersed tunnels

When spotting a site for the construction dock, opportunities to reshape river-banks and coastlines as part of a tunnel construction schememay be observed.For example, the specific tunnel construction costs may be reduced if the projectis associated with a land reclamation scheme.

• Construction processCompared to the boring of a tunnel or the construction of a bridge, much of theimmersed tunnel work is done on shore, in the construction dock. This makesthe process easier to handle for the contractor, thus reducing the uncertaintiesfor construction time and budget.

(ITA 1999)

Even though many advantages for immersed tunnels exist, thefinal choice of construc-tion will always depend on the specific project.

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4EARTHQUAKES

The only load condition considered in the present thesis is the earthquake on the im-mersed tunnel. In this chapter, the basic nature of earthquakes is outlined, and theaffection of the strong ground motion of tunnels is described.

Allthough earthquakes are not very frequent in the region ofthe world were this uni-versity is situated, they have throughout history caused the destruction of countlesscities on nearly every continent. Earthquakes are the leastunderstood of the naturalhazard and in early days were looked upon as supernatural events. The totally unex-pected – nearly instantaneous – devastation of a major earthquake has a unique psy-chological impact which demands serious consideration by society. (Dowrick 1987)

The destructiveness of earthquakes is most recently illustrated to the internationalcommunity with the great consequences of the major earthquake of 12 May 2008in the Sichuan province of China. The casualties count around 70 000, and five mil-lion people are left homeless. In Figure 4.1 some of the damage due to the disaster isshown.

17

Chapter 4. Earthquakes

Figure 4.1: A bank building in Beichuan after the May 2008 Sichuan earthquake. (Commons2008)

4.1 Plate tectonics

An earthquake is a spasm of ground shaking caused by a sudden release of energy inthe earth’s lithosphere (i.e. the crust plus part of the upper mantle). The underlyingcauses of earthquakes are closely related to the global tectonic processes, which arecontinually producing mountain ranges and ocean trenches at the earth’s surface. Themajor tectonic plates are depicted in Figure 4.2. (Clough & Penzien 1975, pp522-525)(Dowrick 1987, pp4-6)

Figure 4.2: The major tectonic plates. The red arrows indicates the movement of the plates.(Commons 2006)

Almost all earthquakes occur at the interface between two plates. The movements ofthe tectonic plates, shown in Figure 4.2 with arrows, are caused by convection in themantle, shown in Figure 4.3. Where plates spread from each other, a ridge is formed,

18

Wave types

and where plates overlap a subduction zone is formed, where the heavier crust platesubducts under the lighter. This is depicted in Figure 4.3. It can be seen in Figure 4.2that the ridges are mostly formed on the oceanic floor.

Outer core

Asthenosphere

Inner core

Spreading ridge

Mantle

LithosphereSubduction zoneSubduction zone

Figure 4.3: The parts of the earth with spreading ridges and subduction zones. After(Com-mons 2007).

If the movement of a plate is obstructed by the neighbouring plate, the friction energywill be saved up, in some cases for decades or even centuries,until the energy isreleased spasmodically as seismic deformation; an earthquake. The rupture planeis called afault, and can in some cases be observed directly on the ground surface,especially in larger shallower earthquakes. (Dowrick 1987, p5) (Kramer 1996, p27)

4.2 Wave types

As the earthquake energy is released along the fault, it propagates through the soil as anumber of waves, which have different characteristics. Thenature of these waves areoutlined in this section, which is based upon Andersen (2006, pp2-4). The four mostcommonly observed waves, P-waves, S-waves, R-waves, and L-waves are shortly pre-sented.

The P-wave is denoted theprimary wavesince it is the first wave to arrive at an ob-servation point. The particle motion is pure dilatation, orpressure. The P-wave isillustrated in Figure 4.4a.

The S-wave is denoted thesecondary wavesince it arrives after the P-wave, typicallyhaving a phase velocity of half the P-wave velocity. The particle motion for the S-wave happens as equivoluminal shear. The S-wave does not appear in a fluid, sinceno shear stress can be generated there. The S-wave is depicted in Figure 4.4b. AS-wave consists of two different components, the SV-wave and the SH-wave. The Vand H abbreviates Vertical and Horizontal, and indicates the direction of the particlemotion. The difference is illustrated in Figure 4.4b and Figure 4.5b. In the first ofthese subfigures the SV-wave is shown, whereas the latter shows the L-wave whichacts like a SH-wave on the surface.

The P- and S-wave are jointly referred to asbody waves, since they propagate throughspace. In opposition to this stands thesurface waves, which count the R-wave (Rayleighwave) and the L-wave (Love wave). These are depicted in Figure 4.5 on the followingpage. The Rayleigh wave moves the particles in ellipses justlike ocean waves. How-

19


Figure 4.4: Deformation produced by body waves: (a) P-wave, and (b) SV-wave. The wavespropagate from left towards right. (Kramer 1996, p19)

ever, the particle motion is retrograde near the surface, cf. Figure 4.5. Opposing tothe P- and S-waves, the Rayleigh wave contains both pressureand shear componentsin the displacement field. The particle motion is greater in the vertical than in thehorizontal direction, which is not clearly indicated in Figure 4.5.

The L-wave is, shortly explained, horizontally polarized shear waves (SH-waves)which are bound to the surface like the R-wave, thus creatinghorizontal horizontalmovement of the earth during an earthquake.

Figure 4.5: Deformation produced by surface waves: (a) Rayleigh wave, and (b) Love Wave.The waves propagate from left towards right. (Kramer 1996, p20)

The present analyses

In a dynamic model, all the above stated types of waves shouldin principle be mod-elled and accounted for, as the earthquake motion propagates from the fault to theproject site. In Figure 4.6 on the next page the propagation of the earthquake to theThessaloniki tunnel is sketched. In the present thesis however, it has been chosen tofocus only on SH-waves, to simplify the problem. It is deemedthat the horizontalcomponent of the ground motion is the more dangerous and thisis mainly caused byS-waves, cf. Fardiset al. (2005, p21). Furthermore, it is stated by (Poweret al.1996)that S-waves are typically associated with peak particle accelerations and velocities,and the focus on SH-waves is widely used,e.g.by Anastasopouloset al.(2007). Since

20

Determination of design earthquake

the waves are refracted as they reach the surface due to decreasing soil stiffness, as itis sketched in Figure 4.6, the horizontal polarization is justified.

Decreasing stiffness

Bedrock Fault

Surface

Figure 4.6: Propagation of the earthquake waves to the tunnel.

In Figure 4.6 it is sketched that the waves firstly propagate through the bedrock. Thisis valid since it is assumed that a significant distance exists between the fault andthe project site. The damping in the alluvial soil is much greater than the dampingin bedrock. Thus, all surface waves are disregarded, since it is assumed that theyare damped away before reaching the project site. On the other hand, surface wavepropagates two-dimensionally and body waves three-dimensionally, which makes ge-ometrical damping more significant for body waves than for surface waves. Hence,geometrical damping and material damping are contradictory quantities, the impact ofwhich should be analysed for a final design.

4.3 Determination of design earthquake

When determining the design earthquake many parameters mustbe evaluated, andmany tools are at hand. In this section, some of these are presented.

This analysis is typical a job for skilled seismologists, but the final determinationshould be made in cooperation with the geotechnical and structural engineer, since thedeformation mode of the structure may influence on what is characterized as the moredangerous earthquake motion.

4.3.1 Size of earthquake

When categorizing earthquakes with a single parameter, the size of the earthquake iseither characterized with the intensity or the magnitude.

Intensity

Intensity is a measure of the destructiveness of the earthquake, as evidenced by humanreaction and observed damage. This is in other words a subjective measure, dependenton the eyes of the beholder. For historic earthquakes, this is the only measure available.Several scales are available, including the Modified Mercalli Scale (MM), the Euro-pean Macroseismic Scale (EMS) and the Japan MeteorologicalAgency Scale (JMA).

21


Common for the scales is the representation of the intensitywith Roman numerals.(PIANC 2001, pp129-130)

Magnitude

Magnitude is an instrumental measure of the size of an earthquake. It is related directlyto the energy released, which is independent of the place of observation. Again, sev-eral scales are available, based on the amplitude of seismograph records. The scalesinclude the Richter local magnitudeML, well known amongst laymen through thenine o’clock news. The moment magnitudeMW or surface magnitudeMS are, how-ever, preferred by seismologists. The scales do not significantly differ for magnitudesup to 6. All magnitude scales have in common the representation of the magnitudewith Arabic numerals. The maximum recorded magnitude is about MW = 9.5 (e.g.Chilean earthquake of 1960) (PIANC 2001, p130)

4.3.2 Seismic hazard analysis

To determine the design earthquake, a seismic hazard analysis can be carried out. Theseismic hazards are the physical phenomena associated withan earthquake, which arelikely to produce adverse effects on human activities. The hazards includei.a. groundfailure and liquefaction, but mostly the ground motion is used as measure, since it iscorrelated to the other hazards.

Both deterministic and probabilistic seismic hazard analyses (DSHA and PSHA) canbe performed. In the DSHA, the nearby potential earthquake sources are examined,and with the help of attenuation relationships or microzonation, the potential size ofthe earthquake at the project site is determined. The strongest earthquake is chosen,and a design to resist this earthquake is performed.

For a PSHA, all nearby seismic sources are incorporated in the design. The probabilis-tic distributions of each sources earthquake potential areincorporated, and a design ismade based on a chosen reliability indexβ, i.e. the possibility that the design earth-quake will be exceeded during a particular time period. For further references onseismic hazard analyses see, for example, (Kramer 1996, pp114-118).

Liquefaction

Liquefaction is a most dangerous phenomenon, which is shortly outlined in the follow-ing. Under earthquake loading liquefaction can occur in areas with loose cohesionlesssoils. Also, liquefaction only occurs in saturated soil, and therefore is most commonlyobserved near rivers, bays and other bodies of water (Kramer1996, p5). As the soildeposit is sheared back and forth, the pore water pressure may rise rapidly, even tothe level of the total stresses, thus eliminating the effective stresses. If this occurs, thestrength and the stiffness of the soil are lost altogether. In this condition large groundmovements can occur. The liquefaction condition ends when the pore water over-pressure has drained, thus restoring the effective stresses. In some cases a drain path

22

Code spectra

can evolve through the upper soil layers, spouting sand and water up as “volcanoes”.(PIANC 2001, p9)

The mechanisms of liquefaction are schematically shown in Figure 4.7. The damagemode of liquefaction has not been examined in the present thesis, since the subsoil inthe project area does not contain any cohesionless soils, cf. Table 5.1 on page 28.

Figure 4.7: Mechanism of liquefaction. (PIANC 2001, p10)

4.3.3 Code spectra

Alternatively to seismic hazard analyses, the design earthquake can be determinedwith the use of a code. This is the most common approach to an aseismic designwhen dealing with ordinary structurese.g. like multi-storey buildings. Typically, thedesign seismic action will be given by a response spectrum, the shape and magnitudeof which are altered according to the ground conditions and geographical region. Twoexamples of such code spectra are shown in Figure 4.8.

I

d

A

)T(

q

0

1

0.25

0 T (sec)

0 0.2 0.4 0.6 0.8 1 1.2 2 3

(a) Greek code (b) Eurocode

Figure 4.8: Examples of earthquake response spectra. The shown parameters are used to ad-just the spectra to the local seismic conditions. (EAK 2000, p18) (EN 1998-12003, p25)

The calculation of a response spectrum is done with a single degree of freedom (SDOF)system, which is subjected to an earthquake. For each distinct frequency, the response

23


spectrum value is the maximal response for the SDOF system, which is tuned so thatthe eigenfrequency is the distinct frequency. The concept is illustrated in Figure 4.9,where the SDOFs with varying eigenfrequencies are shown between the input motionand the response spectrum.

Figure 4.9: The generation of a response spectrum. (Kramer 1996, p571)

Response spectra are directly applicable for the design of conventional buildings inthe frequency domain, but time series are not directly provided from the spectra. Tofind appropriate time series, numerous time series must be analysed, and the timeseries with the best fit to the design response spectrum should be chosen. The motionamplitude can be scaled to make a better fit. In a design process, EN 1998-1 (2003)recommends the use of a minimum of three different time series recordings. In thisthesis however, only a single time series has been used, since the application of moretime series is trivial.

4.3.4 The present thesis

In most cases the design earthquake is not for the structuralor geotechnical consultingengineer to determine. Typically, the design earthquake isprovided in the SpecialConditions of Contract,e.g. in the form of a peak ground acceleration, velocity, anddisplacement, or as a response spectrum.

In the present thesis no analysis has been made to find an appropriate design earth-quake, since the purpose of this thesis has been not to provide a design for the im-mersed tunnel but instead to evaluate the accuracy of the design models. In Section 5.5.1the applied earthquake time series from theMs = 6.2 Aegion 1995 earthquake is pre-sented. This has been chosen because it was at hand, and due tothe geographicalproximity of Thessaloniki and Aegion, shown in Figure 5.5 onpage 33.

Further information on the detailed seismic environment ofThessaloniki is providedby e.g.Pitilakis et al. (2007) and Anastasiadiset al. (2001).

24

The influence of earthquakes on underground structures

4.4 The influence of earthquakes on undergroundstructures

For a structure placed above the ground surface, the most dangerous earthquake mo-tion parameter is normally the acceleration on the surface.The acceleration is typicallyconverted to an applied force using D’Alembert’s principle, and the seismic design ofthe structure is verified (Nielsen 2004, p33).

Seismic design of underground structures is typically verydifferent, since the iner-tia of the surrounding soil is large relative to the inertia of the structure. This meansthat the dominating parameter is the displacement in the surrounding soil. This hasbeen verified by measurements made by Okamotoet al. (1973). Thus, the inertia ofthe underground structure itself becomes of minor importance. The focus in under-ground seismic design, therefore, is on the free-field deformation of the ground and itsinteraction with the structure, as recommended by (Hashashet al.2001, p252).

4.5 Damage modes of tunnels

An immersed tunnel subjected to earthquake-induced strongground motion will de-form in a number of modes at the very same time. In Figure 4.10,the most significantof these modes are depicted. Figures 4.10a and 4.10b show compression in the axialand the cross axial direction, respectively. Figure 4.10c depicts bending of the tunnel,which can be occur both horizontally and vertically, and Figure 4.10d shows sheardeformation of the cross section, denoted asracking.

(a) Compression-extension (b) Compression of tunnel cross section

(c) Longitudinal bending (d) Racking of tunnel section

Figure 4.10: Deformation modes of tunnels due to seismic waves. After Owen & Scholl(1981).

For an immersed tunnel, the critical mode of earthquake induced vibration is the lon-gitudinal oscillations, according to Anastasopouloset al. (2007), since it may lead to

25


decompression of the joint gaskets. This will jeopardize the watertightness and, hence,the safety of the tunnel. Therefore, focus in this thesis is given to the longitudinal de-formation of the gaskets, occuring mostly from mode (a) of Figure 4.10, but also frommode (c). Thus, potential deformation of the cross section of tunnel elements andgaskets is disregarded.

In Section 5.7 on page 38 it is discussed how damage to the tunnel is measured in thisthesis.

4.6 Incoherence

For a structure such as a tunnel that extends over a considerable distance, differentground motions may occur beneath different parts of the structure. This local spa-tial variation of the ground motion is denotedincoherence, and it may exert a veryimportant influence on the response of the structure (Kramer1996, p100).

The incoherence can be caused by a number of factors, three ofwhich are depicted inFigure 4.11. Figure 4.11a show the wave-passage effect, where an inclined wavefrontcauses the motion in locations 1, 2 and 3 to be shifted in time.Figure 4.11b show theextended source effect, where multiple faults generate earthquake waves which willreach the observation points at different times. Finally, the effect of soil heterogeneityis depicted in Figure 4.11c, where inhomogeneities in the soil cause reflection andrefraction of the waves, thus altering the displacements inlocations 1, 2 and 3.

(a) Wave-passage effect (b) Extended source (c) Soil heterogeneity

Figure 4.11: Incoherence. (Kramer 1996, p101)

In the present thesis, only the wave-passage effect is used as the cause of incoherence.While the two other effects could also have been incorporatedwithout disproportionatecosts, this is omitted to simplify the analyses.

The propagation velocity of the earthquake is in principle the velocity of the wavesin the bedrock, stated in Table 5.2 on page 29. Since the three-dimensional wavepropagation in the bedrock is not well accounted for, and since both the velocity of P-and S-waves will influence on the observed surface propagation velocity, it is chosento use anapparent velocitywhich is based on empirical measurements. Typically,apparent wave passage velocities range between 1000m

s- 2500m

saccording to Vrettos

et al. (2007). The apparent velocity is further discussed in Section 5.5.2.

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5DESIGN BASIS

This chapter encloses the basic informations about the projects required for the furtheranalyses.

For the analyses, a lot of parameters have to be determined. These includei.a. the soilparameters, soil stratigraphy, earthquake parameters, aswell as the physical geometry.

In a conventional static analysis, a characteristic value should be determined,e.g.asa five-percent quantile of the strength of a material. After application ofe.g.partialsafety factors, the provided design values should make surethat the design can be ver-ified to be on thesafe side. In a static analysis, it is mostly a trivial task to determine,whether or not a parameter is determined on the safe side. This, however, is not triv-ial in general for a dynamic analysis, and in particular for the analyses in the presentthesis. It ise.g.not easy offhand to say, whether an increase in the shear stiffness in asoil layer will increase of decrease the displacement in thetunnel gaskets. This coulde.g.depend on which eigenmodes of the tunnel are excited.

Hence, as it is not fruitful to determine characteristic anddesign values, in this chapter,best estimates of the “correct” mean value are searched for.Thereafter, in Part 3, thesensitivity of some of the parameters is analysed,i.e. it is analysed how the damage tothe tunnel is affected by changes of a given parameter. This is done in order to examinewhether the determination of a given parameter should be given great consideration,or if a reasonable estimate is sufficient for a final design.

5.1 Geometry of tunnel

The longitudinal section of the immersed tunnel is sketchedin Figure 5.1. The im-mersed tunnel consists of eight tunnel elements, each approximately153m long (COWI2007). The cross section is depicted in Figure 5.2 on the following page.

27

Chapter 5. Design Basis

West East

Existing seabed profile at centreline of tunnel

Upfill

A

B

C

D

10 m

3 m

10 m

Figure 5.1: Longitudinal section of the tunnel. The letters indicate soil layers. After COWI(2007).

Seaward sideShore side Existing seabed level

Dredged trenchLocking fill

12750mm gravel bed

1000mm filter layer1000mm rock protection

34500mm

8700mm

Figure 5.2: Typical cross section of the tunnel. After COWI (2007).

5.2 Geotechnical parameters

As indicated in Figure 5.1, the subsoil in the project area can be divided into four dis-tinct layers. The fill layer is disregarded. In this section,the soil parameters associatedwith these layers are presented.

The four layers are listed in Table 5.1 together with the bulkweights. The thickness ofthe two topmost layers are indicated in Figure 5.1. The thickness of the red clay, layerC, is estimated to 100m to 150m. (COWI 2007)

Table 5.1: The layers of Figure 5.1. (COWI 2007)

Layer Description γ[

kNm3

]

A Loose sandy silty clay and silty clayey sand withoccasional gravel

19

B Medium dense silty clayey sand and firm sandysilty clay with some gravel

21

C Firm red sandy silty clay with little gravel 21D Bedrock -

In the microzonation report of Anastasiadiset al. (2001) dynamic soil parameters ispresented. These include S- and P-wave velocities and quality factors for nine generalsoil layers present in the vicinity of Thessaloniki. The stratification and soil parametersare achieved as the result of a large-scale geophysical and geotechnical survey, wherea detailed geotechnical map has been made. The geotechnicaldata comprised 440boreholes with more than 4000 soil samples and 171 CPTs.

The correlation between the layers in the present interest and the layers of Anastasiadiset al. (2001) is presented in Table 5.2 on the facing page. The comparison has beenmade based on the geotechnical descriptions and bulk densities.

28

C

A

B

C

D

Damping

Table 5.2: Quality factorsQS and velocitiesc for P- and S-waves in the soil. The equivalentlayer is the layer name cf. Anastasiadiset al. (2001). The values in brackets specifythe expectation values. (Anastasiadiset al.2001, p2620)

Layer Equivalent layer cS

[

ms

]

cP

[

ms

]

QS [−]

A B2 200–300 (250) 1800 20-25 (20)B B1 300–400 (350) 1900 15-20 (20)C E 350–700 (600) 2000 6-30 (30)D G 1750–2200 (2000) 4500 180-200 (200)

5.3 Damping

In soil, energy is dissipated by various mechanisms. Due to their high complexity,these mechanisms can not be modelled explicitly. Therefore, some convenient math-ematical formulation which lumps the various energy lossestogether – a dampingmechanism – must be chosen. (Kramer 1996, p567) In this section, the dampingmechanisms used in this project are outlined.

5.3.1 Loss factor

A lot of different quantitative measures of damping exist. The relation between threecommon measures, the quality factor,Q (preferred by seismologists), the dampingratio,ξ, and the loss factor,η, are given as

Q =1

2ξ, η= 2ξ (5.1)

(Kramer 1996, p569).

In the following, the loss factor,η, will be used. For soil, the loss factor normallyranges betweenη= 0.03 to η= 0.05. For the present project, the loss factors have beendetermined from the quality factors in Table 5.2, and are listed in Figure 5.4. For thetunnel, which is cast of concrete, a loss factor ofη = 0.01 is estimated. This is alsoassumed to correspond to the gaskets. The sensitivity of theloss factors are analysedin Section 11.4.1.

5.3.2 Viscous damping

Viscous damping is commonly applied in structural dynamics, and models the be-haviour of a dashpot. The damping is proportional to the velocity, as it can be seen inthe equation of motion for a single-degree-of-freedom (SDOF) system, formulated inthe time domain

k ·u + c · u +m · u = f (5.2)

where a dot () signifies differentiation with respect to time, and wherek, c, m, f andu are the stiffness, damping, and mass coefficients, the load,and the displacement,respectively.

In the frequency domain, it is assumed that the displacementis periodic. This entailsthat the displacement can be written as a linear combinationof harmonic motions,

29


each of which can be expressed asu = eiωt . Differentiation of this expression withrespect to time,t , yields

u = eiωt , u = iωeiωt= iωu, u =−ω2 eiωt

=−ω2u (5.3)

Insertion of these expressions into (5.2) provides

(k + iωc −ω2m)u = f (5.4)

which is the formulation of the equation of motion in the frequency domain. Now, it ischosen to let the damping coefficient be proportional byβ to the stiffness coefficient

c =βk (5.5)

This may be clear for a SDOF system, but it could also be applied for a multi-degree-of-freedom (MDOF) system, where the formulation is a special case of the Rayleighdamping method. Rayleigh damping entails that the damping matrix is written as alinear combination of both the stiffness and the mass matrices.

Now, (5.4) may be written as

(k(1+ iωβ)−ω2m)u = f (5.6)

This leads to the definition of viscous damping as a modification of the stiffness

k∗vis(ω) = k(1+ iωβ) (5.7)

where the star (∗) indicates the modification to incorporate damping. This leaves theequation of motion to be

(k∗vis(ω)−ω2m)u = f (5.8)

Due to linearity, these equations, (5.7)-(5.8) can directly be formulated as matrix equa-tions, to account for the modelling of a MDOF system.

5.3.3 Hysteretic damping

The hysteretic damping model is formulated with the loss factor.

k∗hys(ω) = k(1+ iηsign(ω)) (5.9)

(Andersen 2006, p54)

This damping formulation has been quit widespread in use, mainly due to the verysimple formulation in the frequency domain. Hysteretic damping is frequency inde-pendent (only the sign function enters), which correspondsquite well to the behaviourof soil according to Andersen (2006, p54). Therefore, hysteretic damping is in generala better damping method for soil analysis than viscous damping.

It may, however, be shown that the hysteretic damping model is not causal,i.e. theuse of hysteretic damping may imply that the response appears before the loading isapplied. This effect is, however, insignificant for small loss factors, in the area of whatis normal for soils.

30

Conversion

5.3.4 Conversion

While the hysteretic damping, (5.9), is easily formulated and applied in the frequencydomain, it can not be formulated in time domain;i.e. a formulation compatible with(5.2) can not be made.

Therefore, in this thesis, hysteretic damping will be used for the calculations in thefrequency domain, while viscous damping is applied to the calculations in the timedomain. Conversion between the two damping formulations should be performed in away which yields the most equal output.

By comparison between (5.7) and (5.9), the relation is seen to be

k∗hys = k∗

vis ⇒ iηsign(ω) = iωβ

η= |ω|β (5.10)

This equation, (5.10), entails that in order to correlate viscous and hysteretic damping,it is necessary to choose a frequency at which the damping will be equal. The fre-quency dependence of the two damping mechanisms are sketched in Figure 5.3a. Theconversion frequency should be chosen as the dominant frequency of the system. Thedominant frequency is in the present thesis chosen as the first damped eigenfrequencyof the stratum,f = 1.09Hz. In Figure 5.3b it can be seen that this frequency dominatesthe stratum response to the earthquake.

Damping

ω[

rads

]

hysteretic,iη

viscous,iωβ

Conversion frequency

(a) Relation between viscous and hystereticdamping.

0 0.5 1 1.5

f , [Hz]1.09

(b) The dominant frequency in the responsespectrum in the level of the tunnel. Ex-cerpt from Figure 6.11.

Figure 5.3: Viscous and hysteretic damping.

The relation between the circular frequency,ω[

rads

]

, and the frequency,f[

s−1]

, bothused in Figure 5.3, is

ω= 2π f (5.11)

In Section 11.4.2 it is shown, how the use of either viscous orhysteretic dampingaffects the soil response.

5.4 Design cross section

The actual geometry, as described in Section 5.1 and depicted in Figure 5.1, does notcomply with the simple geometry needed for the domain transformation method de-

31


scribed in Section 6.2. Thus, a simplification is needed. Theirregular layer interfaceshave been reduced to horizontal interfaces located in some mean depth, rounded off toan integer value. Furthermore, the cross section at the middle of the tunnel is analysed,i.e. the water depth is around 10m. Finally, the dynamic parameters are taken as themean values given in Table 5.2. The design section is shown inFigure 5.4.

These assumptions are deemed as realistic for a simple calculation in a real-worldconsulting firm. In Chapter 11 it is analysed how the simplifications may change thephysics of the problem.

z

+0.0

−10.0

−13.0

−23.0

−148.0

Water — Thessaloniki Bay

A Loose clay / sand cS = 250 ms , η= 0.05, γ= 19 kN

m3

B Medium dense sand / firm claycS = 350 ms , η= 0.05, γ= 21 kN

m3

C Firm red clay cS = 600 ms , η= 0.03, γ= 21 kN

m3

Figure 5.4: Sketch of design section for the domain transformation method. The bulk den-sitiesγ are stated in Table 5.1. The velocitiesc and loss factorsη are basedupon Table 5.2 on page 29. The loss factors are determined as the reciprocal valueof the expectation value of the quality factorsQ cf. (5.1) on page 29.

For many ground conditions, the stiffness of the soil – and thus also the shear wavevelocity – will increase with the depth, cf. Anastasopouloset al. (2007, p1070). Thiscould probably also apply for layerC cf. Figure 5.4, but since the goal of the presentthesis is to analyse the presented calculation models and not to provide a final designof the immersed tunnel, this is omitted for the sake of simplicity and transparency.

For the dynamic calculations of the soil response, the waterlevel of Figure 5.4 has notbeen incorporated in the model, since it will not affect the dynamic behaviour of thesoil.

The S-wave velocities in Figure 5.4 should be altered to incorporate damping in ac-cordance with (A.2) and (A.4) on page 130. For layerA, with the gravitational accel-eration set tog = 10 m

s2 , this is done as

µ′= c2

S ·ρ = 118.75MPa

µ=µ′·(

1+ i sign(ω)η)

= (118.75+5.94i)MPa

c∗S =

√

µ

ρ= (250.1+6.2i) m

s

(5.12)

32

C

A

Input motion

where a star(∗) indicates that damping has been incorporated through the complexrepresentation.

5.5 Input motion

In this section, the input motion corresponding to the bedrock is outlined. In Chapter 6it is described how the response on the surface and at the level of the tunnel is calcu-lated.

5.5.1 Time series

In Section 4.3 on page 21 the different methods of determining an appropriate designearthquake are outlined. However, in the present thesis a given time series is simplychosen, since the purpose of this thesis is not to provide a design for the immersedtunnel but instead to evaluate the accuracy of the design models. This generic approachhas made a specific determination of the design earthquake outside the scope of thisthesis.

An acceleration time series from theMs = 6.2 Aegion 1995 earthquake has been de-livered by COWI (1995). This time series has been chosen because it was at hand, anddue to the geographical proximity of Thessaloniki and Aegion, shown in Figure 5.5.

Aegion

Thessaloniki

Figure 5.5: Location of Aegion.

The time series is the horizontal accelerations sampled at arate of 100Hz in an out-cropping bedrock. The acceleration time series is plotted in Figure 5.6 on the nextpage.

As described in Section 4.4, the important earthquake motion parameter for an un-derground structure is the displacements. These are easilyobtained through doubleintegration of Figure 5.6, and are plotted in Figure 5.7 on the following page. It maybe seen that the displacement time series does not end at the starting displacement,u = 0. Thus, a permanent displacement has occurred. This is not compatible withfrequency domain calculations, where the motion must be periodical. Therefore thedisplacement time series has been altered slightly, as it isindicated in Figure 5.7.

33


0 1 2 3 4 5 6 7 8 9 10

−4

−2

0

2

4

Acc

eler

atio

n[

m s2

]

Time, t [s]

Figure 5.6: Acceleration time series. Measured earthquake record from the 1995 Aegionevent.

0 1 2 3 4 5 6 7 8 9 10

−0.08

−0.06

−0.04

−0.02

0

0.02

Time, t [s]

Dis

plac

emen

t,U[m

]

Original dataManipulation

Figure 5.7: Displacement series from 1995 Aegion event. Obtained through double integra-tion of Figure 5.6.

Some of the analyses in this thesis are performed in the frequency domain. With aFourier transformation the displacement amplitude spectrum can be obtained from thetime series, Figure 5.7. It is plotted in Figure 5.8.

0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

Frequency,f [Hz]

Am

plitu

de,A

[

mn

da

ta

]

Figure 5.8: Single-sided displacement amplitude spectrum, obtained through Fouriertrans-formation of Figure 5.7. To obtain a smoother spectrum and a periodic signal, thetime series has been padded with additional zeroes, thus yielding a higher resolu-tion of the Fourier transformation. This is further explained in Section 6.5.2.

34

Apparent velocity

5.5.2 Apparent velocity

As it is outlined in Section 4.6, the incoherence of the earthquake motion is of verysignificant importance for the imposed damage to the tunnel.The apparent velocitydescribes the velocity of the propagating wavefront, and typically fall in the rangebetween 1000m

s- 2500m

s, cf. Section 4.6. For the analyses of the present thesis, the

apparent propagation velocity is set to 1500ms

, since this is the choice of Vrettoset al.(2007), which precise deals with the Thessaloniki immersedtunnel. In Section 12.1the consequences of this choice are analysed through a sensitivity analysis.

5.5.3 Direction of propagation

The wavefront is depicted in Figure 5.9. It propagates with the apparent velocity,and the direction of the propagation is defined with the angleθ. Due to the greatuncertainty associated with the determination of the design earthquake, including thedirection of propagation,θ should be chosen so that the tunnel damage is maximized.

Direction of propagation

θ

Wave frontParticle motion

Tunnel

x

y

z

Figure 5.9: Definition of the direction angleθ.

In the analyses in this thesis, as a point of reference the direction of propagation is setto θ = 45◦. The reason for this is that an oblique direction contains both tunnel axialand cross axial particle motion, as it is illustrated in Figure 5.9. Thus, both compres-sion/extension and longitudinal bending of the tunnel is excitated, corresponding toFigure 4.10a and 4.10c on page 25.

In Section 12.2, the damage to the tunnel is calculated for other direction angles, thusshowing the impact of the choice of angle.

5.6 Stiffness of gaskets

The gaskets joints, which couples the tunnel elements, exhibit highly non-linear be-haviour when deformation in the axial as well in the cross axial direction are applied.In this section the behaviour is discussed, and linear approximations are made.

5.6.1 Longitudinal stiffness

In the present thesis, the Gina gasket profiles has been chosen to type ETS-180-220,which i.a. has been chosen for analysis for the Busan Geoje Fixed Link inSouth Korea(Daewoo 2004). In the longitudinal (axial) direction, the non-linear work curve of thegasket is shown in Figure 5.10. The behaviour of the gaskets during loading is of great

35


importance when analysing the system, since the watertightness of the entire structureis dependent on the compression of the gaskets.

Figure 5.10: Force/compression graph for Gina gasket, type ETS-180-220. (Daewoo 2004)

Initial compression

As described in Section 3.1, the Gina gaskets are compressedduring the installationphase of the immersed tunnel elements. The initial compressive force on the gasketsis determined from the water depth of the tunnel centre, approximatelyd = 16m cf.Section 5.1, and the area of the cross section,Afull = 300.2m2 cf. Appendix D. Thisforce is distributed on the total length of the circumference of the Gina gasket,Lgasket =

84.4m cf. Appendix D, thus yielding a distributed force on the Ginagasket,Finit, of

Finit =γwater ·d · Afull

Lgasket

=10 kN

m3 ·16m ·300.2m2

84.4m

= 569 kNm

(5.13)

The compressive strain can then be read off Figure 5.10 to approximately εinit =

96mm.

Linear approximation

Although the behaviour of the Gina gaskets is highly non-linear, a linear approxima-tion is needed since the present thesis only applies linear analyses, cf. Section 1.3.A possible choice is to take the initial compression as the point of reference, and de-fine the stiffness of the Gina profiles as the tangent stiffness in this point, as shownin Figure 5.11 on the next page. The change in the tangent stiffness, however, is rela-tively significant in this area. The consequences of the approximation are analysed inSection 13.1.

36

Shear stiffness

Figure 5.11: Figure 5.10 modified with a line with an inclination of24MN/m2.

5.6.2 Shear stiffness

The shear stiffness of the gaskets is the resistance againsttransverse and vertical de-formation for the coupling between two adjacent tunnel elements. Two different casesgovern the shear behaviour of the gasket: for small shear displacements, the shearstiffness of the coupling stems from the shear stiffness of the Gina profile itself. Forgreater shear displacements, the stiffness of the couplingstems from shear keys of thetunnel.

Gasket rubber

The shear stiffness of a Gina profile itself depends on the compressive force on thegasket. The higher the compressive force, the higher the shear stiffness. This isi.a.due to the expansion of the gasket cross section when load is applied. (Tonnesen 2008)

A simple approximation of the shear stiffness can be obtained by assuming that thegasket is made of a homogeneous, isotropic and non-compressible material. Thiscorresponds well to be behaviour of rubber. With these assumptions, the shear stiffnessmodulus,G, can be calculated based on a known Young’s modulusE and Poisson’sratioν= 0.5 with e.g.(A.2) to

G =E

2(1+ν)=

E

3(5.14)

Shear keys

If the shear displacements become sufficiently larger,i.e. of an order magnitude ofapproximately 5mm (Tonnesen 2008), the shear keys of the tunnel elements becomeactive. An illustrative sketch of these shear keys is provided in Figure 5.12 on thefollowing page. It can be seen that the shear keys provide resistance to shearing whenthe shear allowance is exceeded, while still allowing deformations in the tunnel axialdirection.

37


Shear allowance

Figure 5.12: Shear keys.

The magnitude of the shear stiffness of the shear keys are deemed to be equal to theone of concrete. How many shear keys there will be in the final structure is unknown.

Linear approximation

Due to the Gina profile and the shear keys, the resulting work curve of coupling be-tween the two adjacent tunnel elements will look something like what is sketchedin Figure 5.13.

Shear allowanceShear displacement

Shear force

Figure 5.13: Work curve of Gina gasket under shear.

As it is easily seen, a linear approximation of Figure 5.13 can not be made with muchdegree of realism intact. Therefore, it is simply defined to use the first branch of thework curve as the stiffness in the general analyses,i.e. the shear stiffness of the gasketsare calculated from the shear stiffness of the profile itself, from (5.14). In Chapter 13it is shown that the choice of gasket shear stiffness has verylittle influence on the finaloutput of the models.

5.7 Damage criterion

The goal of the present analyses is to determine to which extent the immersed tunnelsuffers damage from the strong ground motion generated by the earthquake. To mea-sure this damage several damage modes can be observed, as described in Section 4.5.

It is deemed that the earthquake most potentially will damage the joints between thetunnel elements, as it is stated by Anastasopouloset al. (2007). Thus, the main fo-cus of this report will be on the damage mode where the gasketslooses the initial

38

Damage criterion

compressive strain, hereby endangering the watertightness of the tunnel. This loss ofcompression is generated by relative displacement of two adjacent tunnel elements,as sketched in Figure 5.14. If the initial compression of thegaskets, 96mm accord-ing to Section 5.6.1 on page 35, diminishes towards zero, thewatertightness of thestructure is lost.

Figure 5.14: Sketch of two tunnel element ends. For the right tunnel element, the deformedand undeformed (transparent) states are shown, with a considerable (exagger-ated) relative displacement and rotation. The Gina gasket is not shown.The col-ors indicate which corners that are related, when calculating the gasket damage.

The deformation of the gaskets is, in the Winkler model and the continuum model,calculated in the four corners of each gasket, since the maximum and minimum dis-placement will be in the corners. These corners are depictedin Figure 5.14. Damageto a gasket is defined as the absolute distance between two matching corners. As aconsequence of this definition of damage, other deformationmodes,e.g.racking de-formation cf. Figure 4.10d, shear displacement of a gasket or tensile stresses in thetunnel concrete, may not be analysed.

If the final design should show too much opening in the gasketsthe design should bealtered. This coulde.g.include a different gasket type, longitudinal displacement keysor division of the tunnel into more elements. This would allow more deformation,limit the deformation or distribute the deformation, respectively.

39

PARTIIM ETHODS OF ANALYSES

41

CH

AP

TE

R

6WAVE PROPAGATION THROUGH SOIL

The input to the analysis of the immersed tunnel is a time series of strong groundmotion at different levels of the subsoil. As it is describedin Section 5.5.1, the dis-placement time series from the earthquake records applies only to the level of thebedrock. In this chapter, it is outlined how the time-varying displacements can becalculated above the bedrock, including in the level at the tunnel and on the surface.

The steps of the calculation are illustrated in Figure 6.1. The transformation of theearthquake motion is performed in the frequency domain, andthe transformation be-tween the time and the frequency domain is obtained with Fourier transformations.The more demanding part of the calculation is to establish the frequency responsefunction,H(ω), which couples the input and output spectra.

Bedrock timeseries

FFTBedrock spectrum

H(ω)

Output spectrum

IFFTOutput timeseries

Figure 6.1: The calculation procedure from a time series at bedrock to an output time seriesat a chosen level. FFT stands for Fourier transformation with the Fast FourierTransformation algorithm, and IFFT stands consequently for the Inverse FFTalgorithm.H(ω) is the frequency response function.

In the present chapter, firstly, a generalized version of a geometry is given. The geo-metry is limited to astratum, a horizontally layered soil. The geometry, and thus alsothe computation, is entirely one-dimensional; the only dimension regarded being thedepth. Subsequently, two methods are presented: the semi-analyticalDomain Trans-formation Method(DTM) and an application of theFinite Element Method(FEM).

The calculations in the frequency domain are performed by calculating the responsefor many discrete frequencies, and constructing the frequency response function by

43

Chapter 6. Wave propagation through soil

use of the principle of superposition. This entails that only linear material models canbe used, as it is also discussed in Section 1.3.

6.1 Generalized geometry

A soil, modelled as a stratum, subjected to a forced horizontal displacement at bedrocklevel is analysed. Only vertically propagating SH-waves are modelled, as it is dis-cussed in Section 4.2.

A stratum with J soil layers is examined. A cartesian system of coordinates are in-serted in the top of the stratum, and below the stratum bedrock is modelled as a rigidinterface. A definition sketch of the observed domain is shown in Figure 6.2.

Layer 1

Layer 2

Layer j

Layer J

x1

x2

z1

z2

z j

z J

x3

z

Figure 6.2: General geometry used for the domain transformation and the finite element meth-ods. The planes illustrate the interfaces which separate the soil layers.

The reduction of a real, three-dimensional geometry to a one-dimensional stratumprovides access to a very simple way of calculating the response, in means of theDTM, which analytical calculates the response; the only numerics involved are thediscretization of the frequency range. If two- or three-dimensional wave propagationwere to be taken into consideration, only numerical methodsshould be considered.

Constitutive model

The soil layers are modelled as homogeneous, isotropic, linear viscoelastic materials.The reasons for this are as follows:

• HomogeneityInside one layer the assumption of homogeneity should be evaluated on the basisof the dimensions of the soil particles and the wavelength ofthe wave. Sincethe wavelength is considered to be several orders of magnitude larger than theparticles, homogeneity is justified.

• IsotropyIn reality most alluvial soils will display slight orthotropic behaviour, but sinceno detailed data are available (and most rarely are) the assumption of isotropyis adopted. This will have no effect in the present case, since only a vertically

44

The domain transformation method

propagating SH-wave is analysed, which entails that the only stiffness parameterof importance is the horizontal shear stiffness.

• Linear viscoelasticitySoil is in general a non-linear elasto-plastic material, but linear elasticity isadopted since the non-linear case presents too large a computational workload.In the most cases, the relatively small strains induced by the earthquake makesthe linearization acceptable. According to Krätzig & Niemann (1996), soil sub-jected to shear strains up toγ= 10−4 may be analysed as a viscoelastic material.

The prefixvisco- indicates that damping is applied to the material model. Variousdamping mechanisms are discussed in Section 5.3, accordingto which the applieddamping mechanism for the present analyses should be hysteretic damping, since theanalyses are carried out in the frequency domain.

6.2 The domain transformation method

In Appendix A, the DTM is derived. The method establishes a direct analytical re-lation between the harmonic varying displacement at bedrock and an arbitrary otherlayer at a certain frequency, expressed in (A.22) as

Uj 0

n = Hj 0n (ω) ·Un (6.1)

whereUj 0

n andUn are the displacements in thej ’th layer and the bedrock, respectively,while H

j 0n (ω) is the frequency dependent response function for thej ’th layer. The

calculated frequency response function is exact for the chosen material model and soilgeometry.

With the DTM, the geometry given in Figure 5.4 on page 32 is elaborated. For frequen-cies ranging from0−20Hz, the frequency response function is depicted in Figure 6.3.It can be observed that significant amplification, e.g.H(1.09Hz) = 43, is occurring atseveral damped eigenmodes.

It should be noted that the frequency response function is a complex number whichincludes both the magnitude and the phase shift of the response. Hence, what is plottedon the ordinate axis in Figure 6.3 is the amplitude of the frequency response function.

0 2 4 6 8 10 12 14 16 18 2010

0

101

102

Frequency,f [Hz]

DT

MF

req.

resp

.fu

nctio

n,∣ ∣

H1

0∣ ∣

[-]

Figure 6.3: The amplitude of the frequency response functionH10 for the top of the topmostlayer, calculated with the DTM. The geometry follows from Figure 5.4.

45


6.3 The finite element method

As a verification of the applicability of the DTM, a finite element approach is made tothe same problem. The method is derived in Appendix B. In stead of a direct analyticalrelation between the displacements in a chosen layer interface and the bedrock, asin the DTM, the stratum is now discretized into a number of soil elements, as it issketched in Figure 6.4.

Layer interface

Layer interface

Surface

Soil layer

......

......

Soil element

z

Node

Figure 6.4: The finite element method.

6.3.1 Elements

It is chosen to use second-order elements for the analysis,i.e. elements whose de-formations are described by a second-order polynomial. Second-order elements willprovide a much better approximation to soil deformation dueto a propagating wave,than will linear elements. As discussed by Semblat & Broist (2000), much greateraccuracy may be obtained at a lower computational cost. The shape functions of asecond-order soil element are depicted in Figure 6.5.

(a) Mode 1 (b) Mode 2 (c) Mode 3

Figure 6.5: Shape functions for a second-order soil element. (Jensen & Jørgensen 2006)

46

Frequency response

6.3.2 Frequency response

In Figure 6.6 the frequency response function is plotted forthe same frequency rangeas Figure 6.3 on page 45. It can be seen that the two methods arecapable of producingvery similar output from the same input; only a minor divergence can be observedaroundf = 20Hz. This is considered as a verification of the calculation methods.

0 2 4 6 8 10 12 14 16 18 2010

0

101

102

Frequency,f [Hz]FE

MF

requ

ency

resp

onse

func

tion,| H| [

- ]

Figure 6.6: The amplitude of the frequency response functionH for the top of the topmostlayer, calculated with the FEM, with 10 elements in each soil layer. The geometryfollows from Figure 5.4.

6.3.3 Convergence

The frequency response function depicted in Figure 6.6 has been calculated with 10elements in each soil layer,i.e. a total of 30 elements. While the domain transfor-mation method is exact for each frequency, the accuracy of the finite element methoddepends on the discretization of the domain.

The convergence of the finite element method is plotted in Figure 6.7 for six chosenfrequencies. It can be observed that the number of elements needed to obtain conver-gence raises together with the frequency. The reason for this should be obvious, sincecomplexity of the soil deformation profile also raises together with the frequency, andthus, more elements are needed to reproduce the deformationmode.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

1 Hz3 Hz5 Hz10 Hz15 Hz20 Hz

Number of elements in each layer

Nor

mal

ized

freq

uenc

yre

spon

se

Figure 6.7: Convergence of the frequency response for the finite element method.

47


In Figure 6.7, it can further be seen that with a discretization of around 15 elementsper layer, the response to displacements forced at 20Hz can be accurately calculated.This relatively small number of elements needed, is due to the use of second-orderelements, which are able to reproduce the sinus-like soil deformation, cf. Figure 6.9a,well.

6.3.4 Choice of calculation method

As explained in the present and the preceding section, the DTM and FEM yields com-parable results. The domain transformation method, however, is much more computa-tional efficient, since no excess discretization of the domain is needed. Therefore, thedomain transformation method will be used henceforward in this thesis.

6.4 Frequency response

The frequency response function varies down through the stratum. For the top of thelayers the functions are plotted in Figure 6.8.

0 2 4 6 8 10 12 14 16 18 2010

−1

100

101

102

Frequency,f [Hz]

Fre

quen

cyre

spon

sefu

nctio

n,| H| [

- ]

LayerA , H 10

LayerB, H 20

LayerC, H 30

Figure 6.8: The frequency response functionH for the top of the three layers in Figure 5.4.The black line is identical to Figure 6.3.

Most noteworthy in Figure 6.8 is probably the suppression ofthe response for the topof layer C between frequencies from 6 to 9Hz. In this region the mode of the soilmakes the interface between layerB andC experience very little displacement,i.e. anode is formed. In Figure 6.9 on the next page the response at two time steps is plottedfor three frequencies, and it can be seen that for a frequencyof f = 6.7Hz, there is verylittle response at the interface between layerB andC. Hence, what offhand could looklike a possible computational error, here does has physicalmeaning.

48

C

B

C

B

C

Earthquake response

−30 −20 −10 0 10 20 30

0

20

40

60

80

100

120

f = 1.1Hz

f = 6.7Hz

f = 15.3Hz

Response[-]

Dep

th,z

[m]

(a) θ = 0

−30 −20 −10 0 10 20 30

f = 1.1Hz

f = 6.7Hz

f = 15.3Hz

Response[-]

(b) θ =π/2

Figure 6.9: Response through soil for three frequencies. The harmonic motion is generated bya sine function, dependent on the amplitude and phase shift of the motion.Hence,the response at bedrock,z = 138m, in (a) is A · sin(θ+ψ) = 1 · sin(0) = 0. Thehorizontal lines illustrate the layer interfaces. An animated version of the figure isprovided on the attached DVD.

6.5 Earthquake response

The frequency response function, however, is only interesting when compared to thestrong ground motion generated by an earthquake. Figure 6.10 gathers the earthquakeground motion spectrum and the frequency response functionfrom Figure 5.8 andFigure 6.3, respectively. It can be seen that the first eigenmode of the stratum islocated outside the dominant frequencies of the earthquakedisplacement spectrum.This indicates that only relatively little resonance will occur in the stratum.

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Frequency,f [Hz]

Nor

mal

ized

para

met

ers Earthquake spectrum, Aegion

Frequency response function

Figure 6.10: Comparison between the graphs of Figure 5.8 and Figure 6.3. The graphs havebeen normalized with respect to the maximum value.

6.5.1 Frequency domain

The order of the calculation follows from Figure 6.1 on page 43. The response ofthe soil to the earthquake motion is calculated in the frequency domain according to(6.1). The resulting output spectrum is depicted in Figure 6.11 on the following page,

49


together with the similar spectrum for the mean tunnel level, at 4.85 m depth. It canbe seen that there are no significant differences on the spectra.

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

Frequency,f [Hz]

Am

plitu

de,A

[

mn

da

ta

]

Spectrum at surfaceSpectrum in tunnel level

Figure 6.11: Response spectra. The spectrum for the surface is obtained through multiplica-tion of the two graphs of Figure 6.10.

6.5.2 Time domain

Through an inverse Fourier transformation the response in the time domain can becalculated, cf. Figure 6.1. This is depicted in Figure 6.12.As in Figure 6.11, it canbe seen that there is no significant difference on the response at the surface and at thelevel of the tunnel.

0 5 10 15 20 25

−0.15

−0.1

−0.05

0

0.05

Time, t [s]

Dis

plac

emen

t, [m]

Earthquake motion, AegionResponse at surfaceResponse in tunnel level

Figure 6.12: Strong ground motion and response in upper layer in time domain. The tunnellevel is at4.85m depth.

It should be noted that even though the analysis of the wave propagation has beencarried out in the frequency domain, it yields a result in thetime domain which looksvery physical plausible. This could to some be a little surprising, since the underly-ing assumption for a frequency-domain analysis is that the motion is periodically –something which definitely is not fulfilled in the present case, as can clearly be seenin Figure 6.12.

To obtain a reasonable level of accuracy, it is necessary to pad the original time serieswith zeroes, as it is shown in Figure 6.13b. This imitates a periodic motion, whichcould be be illustrated as in Figure 6.13c.

50

Time domain

(a) Original time series (b) Time series padded with zeroes

(c) “Periodic” motion

Figure 6.13: Padding of the time signal with zeroes before the Fourier Transformationto ob-tain a ”periodic” motion.

Four time series with different amount of padding are plotted in Figure 6.14, fromwhich the necessity of the padding should be obviously. If aninsufficient size ofpadding is used, it can be observed that the output time series will take values differentfrom zero,beforethe input time series begins, as it is clearly seen in Figure 6.14b.Furthermore, if no padding is applied (Figure 6.14a), it is not possible to calculate theresponse after the earthquake time series has ended, because no more information isavailable. In the further analyses, a padding of 150s has been applied, which yieldsthe output of Figure 6.12.

0 5 10 15 20 25−0.15

−0.1

−0.05

0

0.05

y

Time, t [s](a) No padding

0 5 10 15 20 25

−0.15

−0.1

−0.05

0

0.05

y

Time, t [s](b) 10s padding

0 5 10 15 20 25

−0.15

−0.1

−0.05

0

0.05

y

Time, t [s](c) 30s padding

0 5 10 15 20 25

−0.15

−0.1

−0.05

0

0.05

y

Time, t [s](d) 70s padding

Figure 6.14: The effect on the output time series of different paddings. The black line is theinput time series, the red line is the output time series.

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7CLOSED FORM SOLUTION

As a first-order estimate of the deformation of the tunnel, a simple, closed-form solu-tion is adopted. The great advantage of the closed-form solution is the minimal input.This makes it very easy to obtain an estimate on the order of magnitude of the struc-ture’s anticipated deformation. This can be used for initial design considerations aswell as for design verification.

The closed form solution assumes that the deformation of thetunnel is equal to theso-calledfree-fielddeformation. Free-field deformations are the ground strains causedby the earthquake, when the tunnel is disregarded,i.e. all soil-structure interaction isignored. Whether the tunnel deformation is over- or underestimated depends on thestiffness of the tunnel relative to the stiffness of the soil. (Hashashet al.2001, p262)

The free wave field is assumed to consist of the same amplitudes at all locations,differing only with a time shift. The input motion is not a time series, but only themaximal acceleration and velocity of the earthquake. This should be calculated at thelevel of the tunnel, cf. Chapter 6.

7.1 Axial strain

It may be shown that the axial strain,εaxial, due to a propagating S-wave with apparentvelocityCS , may, as stated by Poweret al. (1996), be calculated as

εaxial =vS

CSsinφcosφ+ r

aS

CS2

cos3φ (7.1)

wherevS andaS are the peak particle velocity and acceleration, respectively, r is thehalf width of the tunnel andφ is the angle of incidence of the wave with respect to thetunnel axis.

53

Chapter 7. Closed form solution

(7.1) is derived from the normal strain and the curvature of the free-field deformation,given in Hashashet al. (2001, 264). These are combined with simple beam theory toobtain (7.1). The first term in (7.1) represents the peak axial strain due to soil strainin the axial direction, while the second term is the axial strains due to bending of thetunnel.

7.2 Deformation at gaskets

While the axial strain computed by (7.1) assumes a uniform tunnel cross section, thereal tunnel consists of elements connected by gaskets. As a crude approximation itis assumed that the computed axial strain occurs simultaneously over an entire beamelement. Furthermore, the tunnel is assumed to be infinitelystiff, which should be fairwhen the tunnel is compared to the gaskets. This means that the strain over an entirebeam element can be lumped in the gaskets. Thus, the maximal axial deformation at agasket,∆u, can be calculated from the element lengthle = 153m

∆u = εaxial · le (7.2)

7.3 Input

The peak particle velocity and acceleration enters in (7.1), and should be calculated inthe level of the tunnel. The conversion from displacements to velocities and accelera-tions is performed in the frequency domain, since a double differentiation of a discretetime series in the time domain will generate much numerical noise, so that the outputwill be contaminated severely.

Since the motion is assumed to be periodical, differentiation in the time domain issimply performed by multiplying the signal withiω, i.e. double differentiation isobtained by multiplication with−ω2. This is similar to the calculations performedin Section 5.3.2 on page 29. The displacement spectrum at thelevel of the tunnel isdepicted in Figure 6.11 on page 50.

The time series for the velocity is plotted in Figure 7.1, andthe time series for theacceleration is plotted in Figure 7.2.

0 5 10 15

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time, t [s]

Velo

city

,v[

m s

]

Figure 7.1: Velocity time series for the tunnel level.vmin =−0.86 ms , vmax = 0.67 m

s .

54

Input

0 5 10 15

−4

−2

0

2

4

6

8

Time, t [s]

Acc

eler

atio

n,a

[

m s2

]

Figure 7.2: Acceleration time series for the tunnel level.amin =−5.74 ms2 , amax = 8.01 m

s2 .

The results from the closed form solution are calculated together with the results fromthe Winkler model and the continuum model in Chapter 10.

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8W INKLER MODEL

A widely used model, when analysing immersed tunnels subjected to earthquake load-ing, is the Winkler model. This is usede.g.by Vrettoset al.(2007) and Anastasopouloset al.(2007), and has been used for the design of several immersed tunnels (Kiyomiya1995, p469).

In the Winkler spring model, the soil is represented with independent springs inter-acting with the tunnel which is considered as a beam (Dowrick1987, p243). This isillustrated in Figure 8.1, where the longitudinal, transverse and vertical springs areshown. The Winkler model has been implemented in a Finite Element program codedin MATLAB .

x

y

z

Gasket

Tunnel beam

Tunnel beam

Figure 8.1: Sketch of the Winkler model. A gasket (red line) is showed in greater detailinFigure 8.11.

In the present application of the Winkler model, the tunnel elements are modelled withbeam finite elements with appropriate cross-sectional parameters, while the Gina gas-kets are modelled with multiple springs. The modelling and behaviour of the gaskets

57

Chapter 8. Winkler model

are discussed further in Chapter 13.

The present application of a Winkler foundation for the beamhas been performedwith simple spring finite elements. This entails that the distribution of the springsis discretized to a finite number, according to the degrees offreedom for the tunnelbeams. It is also possible to formulate special finite elements which model a contin-uum distribution of springs. The discretized and continuous fundations are illustratedin Figure 8.2.

(a) Discrete distribution (b) Continuous distribution

Figure 8.2: Winkler foundation.

The simple discrete soil spring distribution has been chosen since it is deemed that withan appropriate discretization, the simple modelling will be sufficiently accurate. Theinternal stress in the tunnel beam will be affected by the discretization, but the stressesare unimportant for the present analysis. The need for an appropriate discretizationcan be realised by a study of Figure 8.3, where it can be seen that the discretizationshould be determined with respect to the frequency and the propagation velocity of theearthquake. Convergence analyses have been carried out in Section 8.2.4 to ensure asufficient discretization.

(a) Sufficient discretization (b) Insufficient discretization

Figure 8.3: The Winkler model with discretized soil springs, subjected to (a) low frequencymotion and (b) high frequency motion. The hatched lines at the bottom are theinput displacement field.

8.1 Model assumptions

The Winkler model is by no means an exact representation of the physical problem.Amongst the assumptions made are:

• The springs on the tunnel are totally decoupled,i.e.no retroaction is possible.• The deformation of the tunnel is limited to that of a Bernoulli-Euler Beam,i.e.

no shear deformation is possible.• The gaskets are approximated with linear springs.• The propagation of the waves from bedrock to the tunnel is assumed to be one-

dimensional.

These assumptions will be analysed through this report by means of a continuummodel, which is described in Chapter 9.

Furthermore, both the Winkler and the continuum model is restricted to some generalassumptions which are:

58

Modelling

• The analysis is entirely linear, as it is discussed in Section 1.3.– The material models are all linear elastic– The possible development of a gap between the soil and the tunnel is not

analysed.– The stiffness’s of the Gina gaskets are assumed to be linear.

• The influence of the pre-stressing of the tunnel, discussedin Section 10.4, isneglected.

8.2 Modelling

The model has been coded inMATLAB on the basis of an existing linear FE programfor static analysis, by Stærdahlet al. (2007). The program has been enhanced withnew finite elements and with the ability to perform dynamic calculations. The changesin the program are described in Appendix C. The program files are enclosed on theattached DVD. In Figure 8.4 a screen dump of the simple outputinterface is shown.

0

100

200

300

400

500

600

700

800

900

10000

2040

0

20

40

Figure 8.4: A screendump of theMATLAB model. The soil springs are illustrated with lines,just like the tunnel beam elements. An animated version of the figure is providedon the enclosed DVD.

In the present section, it is described, how stiffness’s of the soil springs and the gasketsare determined. The material properties are described in Chapter 5.

8.2.1 Soil spring stiffness

The soil springs model the soil-structure interaction. An infinite soil spring stiffness,k = ∞, would imply that the tunnel was restricted to the free-fielddeformations ofthe soil, while a zero soil spring stiffness,k = 0, would imply that no contact existedbetween the soil and the tunnel.

The spring stiffness’s of the soil springs shown in Figure 8.1 are calculated with theFinite Element codeABAQUS. A model has been built in the same way as explainedin Chapter 9, however, the tunnel has been modelled as a rigidbody, since it is onlythe stiffness of the soil which should be determined. The mesh is shown in Figure 8.5.ABAQUS has been chosen for the analysis to make the results obtainedwith the Win-kler model comparable to the results from theABAQUS continuum model.

The equivalent soil stiffness’s are determined by applyingloads to the appropriatedfaces of the tunnel, in a static analysis. From the corresponding deformation of the

59


(a) Cross section (b) Isometric view

Figure 8.5: The meshed domain for the soil spring calculations.

tunnel, the soil stiffness can be determined. Since the performed analysis is entirelylinear, the work curve is straight. Thus, only a single load-deformation relation isneeded to determine the stiffness for a spring.

Transverse and vertical soil springs

The boundary conditions for the determination of the transverse and vertical soil springstiffness’s are:encastre(fully fixed) on the sides and the bottom, and no deformationin the x-axial direction on the ends. The definition of the surface terminology of themodel is given in Figure 8.6. These boundary conditions yield a state of plain strainin the model.

Side

Side

Bottom

End

End

x

y

z

Figure 8.6: Terminology definition for theABAQUS analysis.

The deformation modes are depicted in Figure 8.7 and the calculated soil stiffness’sare given in Table 8.1 on page 62.

Since the transverse and vertical soil springs are determined by plane strain analyses,it is possible to verify the calculated stiffness’s directly in a two-dimensional analyses.This has been done inPLAXIS, a commercial finite element code for soil and rockanalysis. A model equivalent to theABAQUS continuum model has been established.The domain, meshed with 15-node elements, is shown in Figure8.8a. Transverseand vertical loads have been applied and the corresponding deformation figures aredepicted in Figure 8.8b and 8.8c.

The stiffness’s calculated withABAQUS andPLAXIS are listed in Table 8.1, given asspring stiffness’s per metre in the longitudinal tunnel direction. It can be seen thatonly minor differences between the calculated stiffness’sexist. These differences are

60

Transverse and vertical soil springs

(a) Horizontal load (b) Vertical load

Figure 8.7: Deformation modes for the transverse and vertical soil spring analyses inABAQUS. The intensity (from blue to red) of the colour illustrates the magnitudeof the displacement.

(a) Cross section

A

A

AA

AA

(b) Transverse spring determination

A

A

AA

AA

(c) Vertical spring determination

Figure 8.8: ThePLAXIS plain strain model for calculation of the transverse and the verticalsoil springs. The results are given in Table 8.1.

purely due to differences in the discretization. Thus, the calculated soil stiffness’s areverified. In the further calculations, the stiffness’s calculated withABAQUS are used,since the Winkler model will be compared to theABAQUS continuum model.

61


Table 8.1: The calculated soil stiffness’s fromABAQUS andPLAXIS.

Direction Spring stiffnessk[

Nm·m

]

ABAQUS PLAXIS

Transverse 1.492 ·109 1.477 ·109

Vertical 2.668 ·109 2.712 ·109

Longitudinal soil springs

For the calculation of the longitudinal soil springs, the boundary conditions are changed.The objective is to model the deformation of an infinitely long tunnel,i.e. a unit stiff-ness per metre in the tunnel axial direction. The bottom and the sides are still subjectedto encastre, but on the ends, only deformation in thex-axial direction is allowed, cf.Figure 8.6. These boundary conditions, together with a loadon the tunnel end, intro-duces a state of anti-plane strain in the domain,i.e. only deformation in thex-axialdirection is present.

In Figure 8.9 the deformation mode is shown. It has been verified that no significantdisplacements occur in the direction of they- andz-axes. The soil spring stiffness iscalculated in the same way as the in-plane springs, from the deformation due to anapplied load, to0.746 ·109 N

m·m.

Figure 8.9: TheABAQUS mesh and displacement for the longitudinal soil spring stiffness.

To verify the calculation,e.g.an axisymmetric model could be build, in a programwhich allows deformation with the azimuth angleθ. If the radiusR in the model ismuch greater than the widthW of the tunnel, the curvature of the tunnel approacheszero and an “infinite” tunnel is approximated. Thus, for great values of R

W, the ax-

isymmetric model provides a good approximation of the anti-plane strain problem.The principles of the model are sketched in Figure 8.10. The model has not beeninvestigated further.

The calculated soil spring stiffness’s, which will be used for the Winkler model, aresummarized in Table 8.2.

62

Gasket stiffness

WR ≫W

Figure 8.10: A possible axisymmetric model to calculate the longitudinal stiffness.

Table 8.2: The calculated soil stiffness’s.

Direction Spring stiffnessk[

Nm·m

]

Horizontal 1.492 ·109

Vertical 2.668 ·109

Longitudinal 0.746 ·109

8.2.2 Gasket stiffness

The gaskets are modelled with multiple springs, as it is illustrated in Figure 8.11. Asingle longitudinal spring models the axial stiffness, while two shear springs modelthe stiffness in the transverse and vertical direction.

x

y

z

Tunnel beam

Tunnel beamTransverse shear spring,kgask,trans

Longitudinal spring,kgask,long

Vertical shear spring,kgask,vert

Figure 8.11: Modelling of gasket. Excerpt from Figure 8.1.

The equivalent longitudinal spring stiffness,kgask,long, can be found from the lin-earized gasket stiffness chosen in Section 5.6.1 on page 35.The gasket circumferenceis in Appendix D stated as 84.4m, whereby the spring stiffness can be calculated to

kgask,long = 24 MNm2 ·84.4m

= 2.05 ·109 Nm

(8.1)

The shear stiffness of the gaskets is discussed in Section 5.6.2. It follows from (5.14)that the shear stiffness’skgask,trans andkgask,vert are determined as

kgask,trans = kgask,vert =kgask,long

3

= 0.68 ·109 Nm

(8.2)

63


8.2.3 Damping

The damping of the tunnel and the gaskets is estimated to a loss factor ofη = 0.01,according to Section 5.3.1. In the time domain, viscous damping is applied. Thedominant frequency is set to the first eigenfrequency of the soil, f = 1.09Hz, accordingto Figure 5.3b on page 31.

The relation between the element damping matrices and the element stiffness matrices,β, is found according to (5.10) and (5.11) on page 31 to

0.01 = 2π ·1.09Hz ·β

β= 1.46 ·10−3 (8.3)

In the same way, the damping for the soil springs are found. Itis chosen to use theloss factor of the upper soil layers,η= 0.05, cf. Figure 5.4, which for viscous dampingapproximates toβ= 7.30 ·10−3.

8.2.4 Discretization

The tunnel elements are discretized into a smaller number offinite element beams.Since the soil springs are connected to the tunnel at the endsof each beam element,the number of beams also determines the discretization of the soil-structure interac-tion. To determine the discretization needed, a convergence analysis has been carriedout. The result is shown in Figure 8.12. On the ordinate axis the normalized tunneldamage, defined in Section 5.7, is plotted. It can be seen thatsufficient accuracy isobtained with around 20 beam elements per tunnel element, which has been used inthe analyses.

0 10 20 30 40 50 60 70 800.88

0.9

0.92

0.94

0.96

0.98

1

Number of beam finite elements per tunnel element

Nor

mal

ized

max

.ga

sket

defo

rmat

ion

Figure 8.12: Convergence of the Winkler model.

8.3 Input motion

The input strong motion is the earthquake time series calculated with the domain trans-formation method which is accounted for in Section 6.2. The time series at the level ofthe tunnel is depicted in Figure 6.12 on page 50. The calculated time series is applied

64

Output

to the outer ends of the soil springs, shifted in time to modelthe incoherence of thepropagating wave, caused by the apparent velocity.

8.4 Output

The output of the analysis should be the damage of the gaskets, according to Section 5.7.Offhand, the output from the Winkler model is the displacement in the degrees of free-dom, i.e. translations and rotations at the tunnel ends. Translationand rotation of twoadjacent tunnel elements are sketched in Figure 5.14. The translation and rotation ofthe tunnel ends can be transferred to the absolute three-dimensional location of thegasket corners, from which the deformation in the gasket corners can be found.

Firstly, the three-dimensional coordinate of every cornerpoint (shown in Figure 5.14)is calculated. Then, the gasket deformation can simply be calculated as the change inthe absolute distance between two matching corners. A sketch of a tunnel end, withthe corner nodes and the degrees of freedom depicted, is presented in Figure 8.13.

ux ,θx

uy ,θy

uz ,θz~PCentre axis

Figure 8.13: The degrees of freedom at the end of a tunnel element.~P is the direction vectorto a corner node.

8.4.1 Translations

The translations from the degrees of freedom,ux , uy anduz , are simply added to theoriginal coordinates of the corners.

8.4.2 Rotations

The rotational degrees of freedom,θx , θy andθz , yields an resulting displacementof each of the corner nodes which should be taken into consideration. To calculatethis displacement, the direction vector of a corner,~P , is introduced. It is depicted inFigure 8.13. This vector is rotated according to the degreesof freedom, thus calcu-lating the new corner coordinate. The calculation of the rotation is described moreclosely in Appendix E.

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9CONTINUUM MODEL

As an alternativ to the Winkler model, a full continuum modelhas been established inthe commerciel FE programABAQUS . In this chapter, the modelling is described.

TheABAQUS CAE-files and output databases are enclosed on the attached DVD.

9.1 Parts and meshes

The model is built of three parts: Tunnel elements, gaskets and the soil. Instances ofthese parts are gathered in an assembly.

All parts of the model are built of continuum elements, of thetypes C3D20R (brick)and C3D15 (wedge). The brick elements are used for the major part of the domain,and the wedge elements are used to fill minor regions. The elements are shown inFigure 9.1. The elements are quadratic (second-order) elements, which are suitablefor modelling of wave propagation, as it is also discussed inSection 6.3.1 on page 46.

1

2

4

126

913 10

78

3

1511

5

14

face 4

face 1

face 5

face 2

face 3

(a) Wedge element, type C3D15

1 2

3

78

5

4

6

16

1920

1712

13

9

10

11

18

15

14

face 5face 2

face 4

face 1

face 3

face 6

(b) Brick element, type C3D20R

Figure 9.1: Elements used in theABAQUS analysis. (Simulia 2007)

The most important assumptions made during the modelling ofthe continuum model

67

Chapter 9. Continuum model

are stated in Section 8.1. The assumptions are common to the Winkler and the contin-uum model.

9.1.1 Tunnel

The eight tunnel elements are all identical. A tunnel part isshown in Figure 9.2

(a) Cross section view (b) Isometric view

Figure 9.2: The meshed tunnel element part.

9.1.2 Gaskets

The Gina gaskets connect the tunnel elements. A Gina gasket part is shown in Figure 9.3.It should be noted that the mesh of the gasket is generated such that it conforms wellto the mesh of the tunnel elements. This makes the assembly more accurate, as it isdiscussed in Section 9.4.

Figure 9.3: The meshed Gina gasket.

The gasket is modelled 0.20m deep, and the elements are provided with a materialorientation, to make the use an orthotropic material model possible.

The gaskets are modelled with ordinary continuum finite elements. It has also been in-vestigated, if the special gasket elements provided inABAQUS would provided a bettermodelling. These elements, however, entails a very realistic modelling of the gasketbehaviour, since they require the initial compression of the gaskets, obtained duringthe tunnel installation phase, to be a part of the modelling.Even if this would resultin a more accurate, non-linear, modelling, it would not correspond with the modellingof the Winkler model, thus introducing another possible factor of divergence between

68

Soil

the models. These factors are already deemed to be plenty in number, as it is outlinedin Section 8.1.

9.1.3 Soil

The soil is modelled as a single part, divided into the three layers. A hole to containthe tunnel is included. The dredged trench is modelled, and the parameters of thebackfilling are set equal to the parameters of soilA, cf. Table 5.2 on page 29, as nobetter data for the backfilling exist. The order of magnitudefor the parameters of soilA correspond well toe.g.soaked sand, cf. Andersen (2006, p3).

The soil part is shown in Figure 9.4

(a) Cross section view (b) Isometric view

Figure 9.4: The meshed soil part.

9.1.4 Discretization considerations

The mere name of the finite element method implies that a discretization has to bemade. The accuracy of any finite element model is pinned to thechoice of a suitablemeshing of the domain and to an accurate time integration algorithm.

Mesh coarsness

The coarsness of the mesh has been chosen as a balancing between the calculationtime of the model and the accuracy of the obtained solution.

The size of the elements should be compared to the wave lengthof the propagatingwaves, which depends on the frequency of the waves and the propagation velocity. Theslowest wave velocity is the S-wave velocity cf. Table 5.2. For soil layerC, which hasthe largest elements and comprises the majority of the domain, the S-wave velocity is600m

s.

The frequency of the incident waves of importance is maximum2.0 Hz, cf. Figure 6.11on page 50. Thus, the minimum wave lengthlmin can be calculated as

lmin =600 m

s

2.0s−1= 300m (9.1)

The longest side of an element in the meshed model is approximately 38.5m. Thus, noless than seven second-order elements are available anywhere in the model to model

69

A

A

C


each wave component. This is deemed to be sufficient to obtaina proper accuracy ofthe calculation.

Time integration

When performing the time integration, the size of the time step should be given con-siderations in order to obtain an accurate solution. The time step used for the analysesis ∆t = 0.01s, which is equal to the sampling rate of the earthquake time series.

The time integration is inABAQUS performed with the dynamic, implicit scheme. Themethod is called the Hilber-Hughes-Taylor operator, and the method is unconditionallystable for linear systems; meaning that there is no mathematical limit on the size of thetime increment that can be used (Simulia 2007). Thus, the solution will not “explode”when the time step is increased.

The correspondence between the time step, the element size and the wave speed shouldalso be observed. Ife.g. the element size is too small when compared to the otherparameters, it might happen that a wave will pass through an element, without beingnoticed by the element. The correlation is formulated in theCourant condition

c ∆t

h=C , C ≤ 1 (9.2)

(Andersen 2006, 72) wherec andh represent a characteristic set of wave propagationspeed and element size, andC is theCourant number.

It follows from (9.2) that if the greatest wave speed is around c = 2000 ms

, and thetime step is∆t = 0.01s, the smallest element should be around20m. This is obeyedby the majority of the elements in the domain. The exception is the places wherethe geometry directs a finer discretization,e.g. in soil layerA which is only 3m inthickness. In Section 9.6.3 it is analysed, how changes in the time step affects thecalculated displacements inside the domain.

Computation time

With the chosen discretization, a transient analysis of thefirst 15s of earthquake load-ing and response takes about four hours to complete. The computation is performedon the Department of Civil Engineering’s computer cluster,and runs serial on a singlecomputational node. The programmed input files which executes a calculation on thecluster are enclosed on the attached DVD.

9.2 Material modelling

In this section, the material data needed for the model is presented. The continuumelements are modelled with homogeneous, isotropic, linearelastic materials, as it isalso the assumption for the domain transformation method, which forms the input forthe Winkler model, cf. Section 6.1. The material data neededfor such materials for adynamic analysis are the Young’s modulusE , Poisson’s ratioν and the densityρ. The

70

A

Soil

exception is the Gina gaskets which are modelled with orthotropic elements, makingit possible to control the behaviour of the gasket more accurately.

9.2.1 Soil

The densities for the soil layers are given in Table 5.1 on page 28. The Young’smodulus and the Poisson’s ratio can be found from the inverserelationships of (A.2)

E =µ(3λ+2µ)

λ+µ, ν=

λ

2(λ+µ)(9.3)

(Andersen 2006, p8)

The Lamé constantsλ andµ can be determined from the P- and S-wave velocitiescP

andcS , which are given in Table 5.2 on page 29. The relations are: ((A.4) is reprintedfor convenience)

cP =

√

λ+2µ

ρ, cS =

√

µ

ρ

⇒µ= cS2·ρ , λ= cP

2·ρ−2µ

(9.4)

(St John & Zahrah 1987, p171)

The resulting soil data are presented in Table 9.1 on page 73.

9.2.2 Tunnel

The tunnel elements are cast of reinforced concrete. The Young’s modulus for initialcompression, which is of the present interest, can be determined from a deemed ulti-mate strength offck = 50MPa to E0 = 40GPa, according to DS411 (1999, p24). ThePoisson’s ratio is estimated toν= 0.15.

9.2.3 Gasket

The modelling of the Gina gasket is performed such that it corresponds well to theGina gaskets of the Winkler model, illustrated in Figure 8.11 on page 63. Thus, acontinuum equivalence to linear springs is desired. While this may not be the mostaccurate representation of the actual physical problem, itis comprehensible and shouldbehave similar to the Winkler model. The gasket modelling ofthe Winkler model andthe continuum model are depicted in Figure 9.5 on the following page.

The desired behaviour of the Gina profiles can be modelled with an orthotropic mate-rial, whose material stiffness matrix,D, is computed as

D =

1/E1 −ν21/E2 −ν31/E3 0 0 0

−ν12/E1 1/E2 −ν32/E3 0 0 0

−ν13/E1 −ν23/E2 1/E3 0 0 0

0 0 0 1/G12 0 0

0 0 0 0 1/G13 0

0 0 0 0 0 1/G23

(9.5)

71


Transverse shear springLongitudinal spring

Vertical shear spring

(a) Winkler model

x,1

y ,2

z,3

(b) Continuum model

Figure 9.5: Gasket modelling in the Winkler and the continuum model. Excerpts fromFigure 8.11 and Figure 9.3.

(Simulia 2007)

The constants appearing in (9.5) on the previous page can be determined from thedesired behaviour of the gaskets. The indices are defined by the coordinate systemin Figure 9.5, where it is also seen that the gasket should be decoupled between thetransverse, vertical and longitudinal deformation. This is accomplished by setting thePoisson’s ratiosνi j = 0. Thus, the material stiffness matrix is symmetric, which inanycase always should hold by settingν12 = ν21E1/E2 etc.

The stiffness in the longitudinal direction,E1, is determined on the basis of the gasketstiffness found in (8.1) on page 63. The area of the gasket in Figure 9.3 is 103.12m2

and the thickness is 0.2m, thus yielding the equivalent computational stiffness as

E1 =2.05 ·109 N

m·0.2m

103.12m2

= 3.98 ·106 Nm2

(9.6)

The transverse and vertical stiffness’s,E2 andE3, respectively, are not important, sincethe corresponding faces of the gaskets do not interact with any other faces, and thePoisson’s ratios are set to zero. With the same arguments, the shear stiffness in theaxial direction,G23, is not important either.

The shear stiffness of the gaskets is treated in Section 5.6.2. According to (5.14), theshear stiffness’s of the gaskets in the transverse and vertical direction,G12 andG13,respectively, can be determined as

G12 =G13 =E1

3

= 1.33 ·106 Nm2

(9.7)

This calculation assumes Poisson’s ratioν = 0.5, which is true for a incompressiblematerial which is a good approximation for rubber. Above, Poisson’s ratio was setto zero, but this is only to make decoupling possible inABAQUS. The two values ofPoisson’s ratio therefore are absolutely unrelated.

9.2.4 Damping

The damping mechanism is viscous damping, cf. Section 5.3. The relation betweenthe element damping matrices and the element stiffness matrices are determined simi-

72

Summary

larly to (8.3) on page 64. The consequence of the applicationof viscous damping overhysteretic damping is analysed in Section 11.4.2.

9.2.5 Summary

The material data are summarized in Table 9.1. The data of thegasket are listed inSection 9.2.3.

Table 9.1: Material data for theABAQUS model.

Material Density Young’s modulus Poisson’s ratio Damping

ρ[

kg

m3

]

E [Pa] ν [-] β [-]

Soil layerA 1900 0.35 ·109 0.49 7.30 ·10−3

Soil layerB 2100 0.76 ·109 0.48 7.30 ·10−3

Soil layerC 2100 2.19 ·109 0.45 4.38 ·10−3

Reinforced Concrete 2500 40 ·109 0.15 1.46 ·10−3

9.3 Earthquake loading

The strong ground motion generated by the earthquake is applied in the model asforced displacements. Different ways of applying the displacements to the domaincan be chosen. Some of the methods are sketched in Figure 9.6.

(a) Uniform motion (b) Free sides

(c) DTM (d) Transparent boundary conditions

Node without prescribed displacementsNode with prescribed displacements

Prescribed displacementsNode with transparent boundary condition

(e)Legend

Figure 9.6: Possible boundary conditions for a cross section of the soil domain. Forsimplic-ity, only transverse deformations are shown.

73


In Figure 9.6a, the input time series is applied to all nodes on the outer surfaces.However, it is known that this would prescribe incorrect free-field displacements onthe side, since the free-field deformations are calculated in Chapter 6.

Another possibility is shown in (b), where the earthquake displacement time series isapplied only to the bottom surface. While this approach at first may appear feasible, ittotally disregards the stiffness and the mass of the soil outside the modelled domain,thus providing a too flexible model.

Therefore, the boundary conditions of (c) are applied to thecontinuum model. Onthe sides of the domain, the deformation is prescribed according to the displacementsin the specific level, calculated with the domain transformation method as describedin Section 6.2. Thus, the far-field displacements are prescribed fully by the free-fielddeformations, while the near-tunnel field are left to the continuum model to calculate.

Another possible way of modelling the boundary conditions appropriate could be withtransparent boundary conditions(TBCs), depicted in (d). TBCs do not prescribe anymotion as standard Dirichlet or Neumann conditions (e.g.displacements or stresses,respectively) but are formulated such that they absorb the outgoing energy of thewaves, thereby preventing reflection. TBCs are normally formulated for waves in aspecific direction, but multi-directional formulations have been provided bye.g.Hig-don (1992). However, TBCs do not provide full transmission for domains with freesurfaces, as discussed by Andersenet al. (2007, p47). To absorb the surface waves,e.g.Rayleigh waves, Bambergeret al. (1990) has suggested the use of buffer zones,socalled “ears” which are applied to the model close to the free surface and appliesfictitious damping.

Instead of TBCs,infinite elementscould be applied. This type of artificial boundaryconditions is directly available inABAQUS , however not in theCAE-interface, and areonly fully transmitting in a specific direction (Simulia 2007). Yet another possibilityfor the formulation of a wave-radiating domain could be an application of the boundaryelement (BE) method, which in its formulation have an inherent ability to radiatewaves. BEs could be coupled with FEs, as it is discussed bye.g. Andersenet al.(2007, p54-56).

While the concept of transparent boundary conditions seems commendable, the appli-cation can be quite tortuous, especially if another formulation than infinite elementsshould be implemented in theABAQUS code. However, the use of wave-transmittingboundaries would be essential ife.g. the near-field was prescribed and the far-fieldunknown, as it is the case for the wave motion frome.g.pile driving. In the presentthesis, since the far-field displacements are prescribed and the near-field the area ofinterest, the concept of (c) has been employed.

The incoherence of the wave, due to the apparent velocity, isaccounted for with a timeshift for each node, dependent on the three-dimensional distance from the node to thewave front.

The automated generation of a specific time series for each and every boundary nodeis not possible in the CAD-likeABAQUS CAE interface. Therefore, a user subroutine,disp, has been implemented. The subroutine has been written inFORTRAN and isenclosed on the attached DVD.

74

disp

Assembly

9.4 Assembly

The above mentioned parts are put together in an assembly, a section of which isshown in Figure 9.7.

ZY

X

Figure 9.7: The assembly.

9.4.1 Ties

The surfaces are connected with surface-to-surfaceties, which share the nodes of ad-jacent surfaces. This reduces the size of the system matrices, eliminating the degreesof freedom at the node of the slave surfaces.

This modelling has been used, since it is deemed that there will be no significant slipbetween the tunnel and the soil, nor will there be any longitudinal shear displacement.If this should be the case, a more appropriate modelling can be obtained by use ofsurface-to-surface contact interactions. These are, however, non-linear in nature.

With the use of ties the same fundamental problem with nominal tensile stresses in thesoil, discussed in Section 1.3, arises. However, due to the oscillations around the stateof equilibrium, the use of ties in the present context is justified.

9.4.2 Master and slave surfaces

To establish a tie between two surfaces, a master surface anda slave surface should bedefined. During the solving of the matrix system, the ties then exclude the degrees offreedom at the nodes of the slave surfaces. Instead, the degrees of freedom at nodeson the master surface are used, thus reducing the size of the matrices.

In general, two surfaces connected with ties can not be expected to have nodes withthe very same coordinates. The link between a node on the slave and master surfaceis therefore computed for each slave node, such that each slave node connects withthe nodes of the master surface, which is nearer to the slave node. This is shownin Figure 9.8 on the following page, where slave surface nodec connects to node402 on the master surface. If the projection point of the slave node on the mastersurface does not lie directly onto a master surface node, thedegrees of freedom areinterpolated according to the shape function of the master surface. Hence, the degrees

75

c

402


of freedom e.g. for slave surface nodea are calculated from the degrees of freedom ofmaster surface nodes202, 203, 302 and303.

b

c

a

104

203

204

304

404

504

102

502

103

503

403

402

101201 301

401

501

202 302

303

slave surface nodes

Figure 9.8: A tie constraint between slave nodes and a master surface. (Simulia 2007)

In the special case of the tie between the tunnel and the Gina gasket, it does not makeany difference, which surface is the master or the slave, since the meshed are identical.However, in most cases the choice of master and slave surfaces requires some consid-eration. The slave surface mesh should, in general, be finer than that of the mastersurface; thus making sure that a connection will exist between all nodes of the twosurfaces.

In the present model, the ties between the soil and the tunnelelements are modelledwith the tunnel as slave surface. For the ties of the gaskets and the tunnel elements,the tunnel elements therefore are the master surfaces. Thisis necessary, since no nodecan belong to two slave surfaces.

9.5 Output

The required output is specified in Section 5.7 on page 38. Thecontinuum modelserves the purpose of analysing the model assumptions connected to the Winklermodel. Therefore, the output of the continuum model should be made comparableto the output of the Winkler model, described in Section 8.4.

9.5.1 Deformation at corners

Since the corners of the gaskets are nodes in the continuum model, the displacementscan be extracted and the deformation at the corners can be calculated directly.

9.5.2 Degrees of freedom equivalent to Winkler model

To compare the continuum model to the Winkler model when no gaskets are mod-elled, as it is done in Section 10.2.5, it is necessary to be able to equate the three-

76

a

202

203

302

303

Translational DOF

dimensional displacement field fromABAQUS to the Winkler model, as it is illustratedin Figure 9.9.

Figure 9.9: Sketch of the problem: the conversion from the continuum model (right) tothebeam model (left). The arrows illustrate the degrees of freedom.

The reduction of the continuum model to six degrees of freedom (DOF) in an equiv-alent beam involves a lot of choices, and no perfect solutionexists. Furthermore, allinternal modes of deformation,e.g.warping or racking (cf. Section 4.5), in the con-tinuum model are disregarded.

Firstly, it is chosen to use the corner nodes and the quadrilateral they form as the basisfor the calculation, thus disregarding all other nodes. Thecalculation is implementedin the postprocessing program, coded inMATLAB and enclosed on the attached DVD.

Translational DOF

The translational degrees of freedom are easily calculatedas the simple mean value ofthe displacement for the four corners.

Rotational DOF

The rotational degrees of freedom can not be defined unambiguous from the displace-ment in the four corner, since only three nodes are necessaryto define a plane in thethree-dimensional space. Since only small rotations are present, it is chosen to definethe rotation about an axis as the rotation of the direction vector between two centrepoints, as it is sketched in Figure 9.10 on the next page for the rotation about they-axis. In the depicted example, only the coordinates in thex z-plane are used for thecalculation ofθy .

77


Centre point

Centre point

Vertical axis

θy

Corner nodex

y

z

Figure 9.10: Rotational DOF from corner nodes.

9.6 Verification of models

In this section, the continuum model is used for verificationof the DTM. It is alsoverified that the free-field waves applied in the Winkler model can be reproduced bythe continuum model. Finally, the size of the time step is analysed.

9.6.1 Boundary conditions

Firstly, it is verified that the applied boundary conditions, the time series calculatedwith the DTM, is correct and corresponds to the material properties entered inABAQUS.This is tested by applying the earthquake motion to a layeredsoil domain without thetunnel, depicted in Figure 9.11.

Figure 9.11: The layered soil domain without tunnel. The dots show the nodes for whichthetimeseries is plotted in Figure 9.13.

The Aegion time series is applied to the soil domain in Figure9.11 without any delaydue to apparent velocity. The displacements on the sides of the domain are calculatedwith DTM, using hysteretic damping.

A screen dump of the deformation is shown in Figure 9.12 on thenext page, and twotime series are plotted in Figure 9.13. It should be noted that very little difference

78

Boundaries calculated with viscous damping

exists between the displacement at the outer surface, whichis forced, and the dis-placement inside the domain, which is calculated in abaqus.This verifies both theDTM and the continuum model.

Figure 9.12: The soil domain during the execution of the time series. Note that the defor-mation is uniform over the cross section. An animated version of the figureisprovided on the enclosed DVD.

0 5 10 15

−0.15

−0.1

−0.05

0

0.05

Time, t [s]

Dis

plac

emen

t,u2

[m]

Outer nodeInner node

Figure 9.13: Displacement time series for the two nodes set off in Figure 9.11. InputfromDTM with hysteretic damping.

Boundaries calculated with viscous damping

Even though the time series plotted in Figure 9.13 seem to be very alike, small dif-ferences exist. This is more clearly seen at places with highacceleration. This effectis due to the different damping models used for the calculation of the input forceddisplacements and in the continuum model itself. While the domain transformationmethod uses hysteretic damping in the frequency domain, thetime domain calcula-tion in ABAQUS utilizes viscous, proportional damping. The differences are furtherdescribed in Section 5.3.

To illustrate the consequence of the use of different damping methods, new time se-ries have been generated with the domain transformation method, now using viscousdamping, even though this damping method corresponds less well to the behaviour ofsoil. The corresponding time series for the two nodes shown in Figure 9.11 are plottedin Figure 9.14 on the following page.

To see the difference between Figure 9.13 and Figure 9.14 more clearly, close-ups areplotted in Figure 9.15 on the next page. Here it is clearly seen that if the same dampingmodel is used for the boundary conditions and for the continuum model itself, a muchmore accurate reproduction is possible. However, since hysteretic damping still is

79


0 5 10 15

−0.15

−0.1

−0.05

0

0.05

Time, t [s]

Dis

plac

emen

t,u2

[m]


Figure 9.14: Displacement time series for the two nodes set off in Figure 9.11. InputfromDTM with viscous damping.

regarded a better model of the behaviour of soil, but simply not available in the timedomain, hysteretic damping will remain to be used for the calculations with the DTMhenceforth.

5 5.5 6 6.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time, t [s]

Dis

plac

emen

t,u2

[m]


(a) Hysteretic damping in DTM

5 5.5 6 6.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time, t [s]

Dis

plac

emen

t,u2

[m]


(b) Viscous damping in DTM

Figure 9.15: Close-up of Figure 9.13 and Figure 9.14.

Furthermore, the same difference between the soil responsecalculated with hystereticand viscous damping has been observed in Section 11.4.2 on page 108, as a clearverification of the results presented in Figure 9.15 and a verification of the coding ofthe continuum model.

Incompatible boundaries

It may not be obvious that significant difference could existbetween forced displace-ments at the outer surface of the domain and the calculated displacements inside thedomain. The importance of compatibility between the stiffness and mass of the modeland the forced displacements can however easily be illustrated,e.g.by using the orig-inal Aegion time series for all forced displacements, exactly as it is sketched in Fig-ure 9.6a on page 73. A calculation with these boundary condition yields the displace-ments plotted in Figure 9.16. It is clearly seen that the incompatibility between the

80

Representation of free-field soil deformation

continuum model and the boundary conditions provides greatdifference between ex-citation and response.

0 5 10 15

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time, t [s]

Dis

plac

emen

t,u2

[m]


Figure 9.16: Displacement time series for the two nodes set off in Figure 9.11. Inputat allboundary surfaces is the original earthquake time series, as sketchedin Fig-ure 9.6a,i.e. the prescribed motion is incompatible with the soil in the domain.

9.6.2 Representation of free-field soil deformation

The Winkler model assumes that the wave propagation is one-dimensional, meaningthat the response on a given point on the soil surface can be calculated directly fromthe displacement time series in the bedrock directly below the point. This is illustratedin Figure 9.17a. However, it seems reasonable that the wavesin some cases couldpropagate in two or three dimensions, making the calculation of the surface responsemore complicated. The principle of multi-dimensional wavepropagation is sketchedin 9.17b.

(a) One-dimensional (b) Multi-dimensional

Figure 9.17: Wave propagation.

To examine, how the model performs when an apparent velocityis present, the cal-culations of Figure 9.12 and Figure 9.13 are performed again, but with an apparentvelocity of 1500m

s. The angle of incidence, defined in Section 5.5.2, is set to0◦. The

deformed soil body is depicted in Figure 9.18.

Similar to Section 9.6.1, the displacement time series is plotted for two surface nodes.It can be seen that the time series are very much alike, exceptfor the time shift, causedby the apparent velocity. Some minor differences are present, but not significant, andcould be due to the different damping models, discussed in Section 9.6.1. This indi-cates that for the present analysis,i.e. for a SH-waveshifted with aapparent velocityin astratum, the assumption of one-dimensional wave propagation seemsOK.

81


Figure 9.18: The deformed soil domain fort = 6s. Apparent velocity: 1500ms , angle:0◦. Ananimated version of the figure is provided on the enclosed DVD.

0 5 10 15

−0.15

−0.1

−0.05

0

0.05

Time, t [s]

Dis

plac

emen

t,u2

[m]

Outer nodeOuter node correctedInner node

Figure 9.19: Displacement time series for the two nodes set off in Figure 9.11 when anap-parent velocity of 1500ms is applied. The red line is the outer node correctedfor the time shift caused by apparent velocity. Input from DTM with hystereticdamping.

9.6.3 Time step

The discretization of time should be sufficiently fine to ensure that all important wavescan be modelled, while still keeping computation time at a reasonable level, as it isdiscussed in Section 9.1.4.

To quantify the importance of the time step, the analysis of Section 9.6.2 is performedagain, but with different time steps. As measure of the errormade as consequenceof the discretization, the data plot of Figure 9.19 is used. Correction for the apparentvelocity has been made and the outer and inner node displacements are analysed, cor-responding to the red and the blue line of Figure 9.19. The computational error,e, fora given time step is computed as the root-mean-square (RMS) value of the differencebetween each data point.

e =

√

∑nj=1

(uouter, j −uinner, j )2

n(9.8)

wheren is the number of data points. For four different time steps, the RMS error isplotted in Figure 9.20.

No clear indications of a reduced RMS error for finer time steps can be observed. Thisis interpreted as an indication that convergence with respect to the time step is alreadyobtained. This is supported by the minor size of the error, approximately 2mm.

82

Time step

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.021.8

1.85

1.9

1.95

2

2.05x 10

−3

Timestep,∆t [s]

Roo

tmea

nsq

uare

erro

r,e[m

]

Figure 9.20: Influence of time step in the continuum model of the soil domain.

Even though the time step∆t = 0.01s provides the highest value of the error inFigure 9.20, it is chosen for the further analyses none the less, since convergenceis obtained and since it provides a reasonable calculation time, cf Section 9.1.4. Fur-thermore, the sampling frequency of the input earthquake time series is 100Hz, cor-responding to the chosen time step. Thus, every data point inthe input time series isused for the analysis. This would not be the case if time stepsgreater than∆t = 0.01s

were applied.

83

PARTIIIANALYSES OF METHODS AND

PARAMETERS

85

CH

AP

TE

R

10COMPARISON OF MODELS

In this chapter, the available models are held up against each other. The closed formsolution, the Winkler model and the continuum model are elaborated, and the differ-ences between them are quantified and analysed.

10.1 Initial results from design basis

Firstly, the damage to the tunnel is calculated with the three different methods pre-sented in this thesis. The basis for the calculations is the parameters discussed andpresented in Chapter 5;i.a. an apparent velocity of1500 m

sand an angle of propaga-

tion of 45◦.

As it is described in Section 5.7, the focus of the analysis will be on the openingand compression of the gaskets. Therefore, the maximal opening and compression ofall gaskets are calculated. This will be the important parameters for a design of theimmersed tunnel.

It can however, when comparing models, to some extent be bewildering to deal withmaximal values, since the place of occurrence for the maximal value may vary. There-fore, to prepare for understanding of the model behaviour, atime series for a specificgasket corner is also presented. The chosen corner is depicted in Figure 10.1 on the fol-lowing page. It is the corner of the centre gasket with the highesty andz-coordinate.

10.1.1 Closed form solution

For the closed form solution, the gasket deformation is estimated with the help of (7.1)and (7.2) on page 54. The input motion data is given in Figure 7.1 and Figure 7.2 onpage 55.

87

Chapter 10. Comparison of models

x

y

z

Figure 10.1: The gasket corner chosen for comparative output, emphasized with an orangedot.

The upper value of the deformation of a gasket,∆u, is calculated with the maximumabsolute value of the velocity and acceleration. This is what typically will be appliedin a real-world initial estimate.

∆u = le ·vS

CSsinφcosφ+ r

aS

CS2

cos3φ

= 153m ·0.86 m

s

1500 ms

· sin45◦ ·cos45◦+34.5m

2·

8.01 ms2

(1500 ms

)2·cos3 45◦

= 47.2 ·10−3 m

(10.1)

Due to the very simple approximation of the tunnel, the calculated gasket deformationof 47mm does not correspond to any specific node. Likewise, since theclosed formsolution assumes a simple harmonic motion, the calculated deformation is both theopening and compression of the gasket.

10.1.2 Winkler model

The Winkler model is outlined in Chapter 8 and in the following, the results are given.

In Figure 10.2, the boundaries for the gasket deformation are depicted as function oftime. Positive deformation is opening of the gasket; negative deformation is compres-sion of the gasket. The maximum calculated opening of a gasket corner is25.9mm,while the maximum calculated compression of a gasket corneris −32.5mm.

For the gasket corner shown in Figure 10.1, the deformation time series is shownin Figure 10.3 on the next page. The maximum opening and compression of the corneris 24.6mm and−30.6mm, respectively.

10.1.3 Continuum model

The continuum model is presented in Chapter 9. The deformed domain is depicted inFigure 10.4.

In Figure 10.5, the boundaries for the gasket deformation are depicted as functionof time. The maximum calculated opening of a gasket corner is2.49mm, while themaximum calculated compression of a gasket corner is−2.35mm.

88

Comparison

0 2 4 6 8 10 12 14

−0.03

−0.02

−0.01

0

0.01

0.02

Time, t [s]

Def

orm

atio

n,∆

u[m

]

MaximumMinimum

Figure 10.2: The Winkler model. Boundaries for the gasket corner deformation.

0 5 10 15−0.03

−0.02

−0.01

0

0.01

0.02

Time, t [s]

Def

orm

atio

n,∆

u[m

]

TotalAxial, xTransverse,yVertical, z

Figure 10.3: The Winkler model. Deformation time series for the gasket corner inFigure 10.1.

Figure 10.4: The deformed domain. An animated version of the figure is provided on theenclosed DVD.

For the gasket corner shown in Figure 10.1, the deformation time series is shownin Figure 10.6 on the following page. The maximum opening andcompression of thecorner is2.25mm and−2.11mm, respectively.

10.1.4 Comparison

In Table 10.1 on the next page, the calculated deformations for the standard parametersare listed. Firstly, it is clearly seen that the three modelsyield very different results.

89


0 5 10 15

−2

−1

0

1

2

x 10−3

Time, t [s]

Def

orm

atio

n,∆

u[m

]MaximumMinimum

Figure 10.5: The continuum model. Boundaries for the gasket corner deformation.

0 5 10 15−2

−1

0

1

2

x 10−3

Time, t [s]

Def

orm

atio

n,∆

u[m

]


Figure 10.6: The continuum model. Deformation time series for the gasket corner inFigure 10.1.

The closed form solution provides nearly twice the deformation of the Winkler model,which furthermore calculates deformations of more than 10 times the deformation ofthe continuum model.

Table 10.1:The calculated gasket deformations. All deformations are in mm.

ModelMax. gasket Max. gasket Max. opening Max. compression

opening compression chosen corner chosen corner

Closed form solution 47.2 −47.2 47.2 −47.2

Winkler model 25.9 −32.5 24.6 −30.6

Continuum model 2.5 −2.4 2.3 −2.1

It should come as no surprise that the closed form solution provides the highest valueof the gasket deformation. With this method many assumptions has been made, someof them on the very safe side. For instance, when the inertia of the system is con-sidered, it may seem unreasonable on the safe side to use the extreme values of thevelocity and acceleration as input to the closed form solution. To obtain a more accu-rate estimate, it could be tried to deem “equivalent” valuesof the input parametersvs

andas from Figure 7.1 and Figure 7.2 on page 55. Neither does it seemreasonableto multiply the calculated maximal strain with the tunnel length to obtain a deforma-tion at the gasket, as it is stated in (7.2). However, it is notpossible to deem a more

90

The Winkler model vs. the continuum model

accurate method without further calculations.

When the simple character of the closed form solution is contemplated in relationwith the complex nature of the problem, it should not cause surprise that the resultsdiffer from the more detailed calculations. In fact, the difference from the closed formsolution to the Winkler model may seem minor.

In the further discussion and analysis, the primary focus will be on the differencebetween the Winkler model and the continuum model. The more detailed modelling ofthe continuum model could be expected to provide a differentresult, but that the gasketdeformations differ with an order of magnitude may be characterized astounding.

10.2 The Winkler model vs. the continuum model

The great difference between the gasket deformations calculated with the Winklermodel and the continuum model, shown in Table 10.1, is analysed in the followingsection.

10.2.1 Crude errors

Firstly, it should be verified that no crude errors have been made in the modelling. Tothis end, the displacement time series in a single node are plotted, to verify that thedisplacements inside the models are comparable for the Winkler model and the con-tinuum model. The selected node is located in the gasket corner shown in Figure 10.1.In Figure 10.7, this gasket is depicted, and the selected node is shown. The node islocated onSide 1of the gasket, defined as the the side with the lowerx-coordinate.

x

y

z

Side 1,x1

Side 2,x2 > x1

Figure 10.7: Definitions of sides of the gaskets. The dot marks the node plotted inFigure 10.8.

In Figure 10.8 on the next page, the displacements in the degrees of freedom of theselected node are plotted for the Winkler and the continuum model. It can be seen thatthe displacements are very much alike. This entails that theearthquake displacementsof the Winkler model and the continuum model make the gasket corner translate inmuch the same way; with thesame order of magnitude. Thus, it is rendered probablethat no crude errors related to the wave propagation to the tunnel have been made inthe modelling.

91


Furthermore, in Figure 10.8 the displacements in the vertical degree of freedom shouldbe observed. In the Winkler model, Figure 10.8a, no displacements occur, while mi-nor displacements are present in the continuum model, Figure 10.8b. This differenceis due to the wave propagation modelled with the continuum model. The input dis-placements does not have any vertical component (since a SH-wave is assumed), andin the Winkler model this is directly transferred to the tunnel, while the waves in thecontinuum model are propagated with respect to the three-dimensional domain.

0 5 10 15

−0.1

−0.05

0

0.05

0.1

Time, t [s]

Dis

plac

emen

t,u[m

]

x, axialy , transversez, vertical

(a) Winkler model

0 5 10 15

−0.1

−0.05

0

0.05

0.1

Time, t [s]

Dis

plac

emen

t,u[m

]


(b) The continuum model

Figure 10.8: Displacements in the degrees of freedom at the corner shown in Figure 10.1.

10.2.2 Three-dimensional wave propagation

In the Winkler model the wave propagation is assumed to be one-dimensional, whereasthe waves in the continuum model have the ability to propagate three-dimensionally.The consequence of this is analysed in Section 9.6.2, where it is shown that in thepresent case, this effect does not yield any significant importance in the results.

10.2.3 Deformation in gasket

Apparently, the orders of magnitude of the displacements correspond well to eachother in the Winkler and the continuum model, but the deformations in the gasketsdiffer with an order of magnitude. Therefore, the deformation in the gasket depictedin Figure 10.1 and Figure 10.7 is analysed further.

92

Transverse and vertical deformation

The deformation in the corner of the gasket has been plotted in Figure 10.3 andFigure 10.6. When the deformations in the respective directions are contemplated,it can be seen that in both models, the deformation in the axial direction dominatesmost heavily. Some transverse deformation is calculated inthe continuum model, butthe total three-dimensional deformation depends almost exclusively on the axial defor-mation. This was also to be expected, since deformations perpendicular to the tunnelaxis – due to the geometry – does not contribute very much to the change in distancebetween the corner nodes.

Transverse and vertical deformation

In Figure 10.9, the transverse and vertical deformations ofFigure 10.3 and Figure 10.6are plotted together. It can be seen that all though the calculated displacements in noway are the same, the orders of magnitude correspond well to each other. Hence, themajor difference between the Winkler and the continuum model apparently is to befound in the axial deformation.

0 5 10 15

−1

−0.5

0

0.5

1

x 10−3

Time, t [s]

Def

orm

atio

n,u

[m]

Transverse, WinklerVertical, WinklerTransverse, ContinuumVertical, Continuum

Figure 10.9: Transverse and vertical gasket deformations of Figure 10.3 and Figure 10.6.

Translations and rotations in gasket centre

To examine which tunnel deformation mode causes the axial gasket deformation inthe two models, the gasket deformation is now expressed in terms of the degrees offreedom for the tunnel beam,i.e. rotations and translations in the gasket centre. Forthe Winkler model, the degrees of freedom are given as the direct output, while theymust be calculated for the continuum model. The problem is sketched in Figure 9.9on page 77, and the calculation is outlined in Section 9.5.2.

The deformations are calculated as the difference in the displacements of the degreesof freedom between side 1 and side 2 of the gasket, cf. Figure 10.7. The translationaldegrees of freedom are plotted in Figure 10.10 on the next page, while the rotationaldegrees of freedom are plotted in Figure 10.11 on page 95. Forthe rotational degreesof freedom, the rotations are converted into displacementsin the tunnel axial direction.Hence, the rotation about thex-axis, socalledroll , is not plotted, since this does notprovidex-axial displacement.

93


0 5 10 15−0.03

−0.02

−0.01

0

0.01

0.02

Time, t [s]

Def

orm

atio

n,∆

u[m

]


(a) Winkler model

0 5 10 15

−2

−1

0

1

2

x 10−3

Time, t [s]

Def

orm

atio

n,∆

u[m

]


(b) Continuum model

Figure 10.10:Translational deformation for the gasket centre.

By studies of Figure 10.10 and Figure 10.11, it is again seen that thex-axial trans-lational degree of freedom provides nearly all the deformation of the gasket. Thisis common to the two models. Furthermore, in both models the rotation about thevertical axis yields only approximately 10% of the axial deformation. Thus, only ax-ial deformation andnot bending of the tunnel governs the gasket damage during thepresent earthquake.

Gasket approximation

The Gina gasket in the Winkler model is represented with three springs, as it issketched in Figure 8.11 on page 63. Hence, no rotational stiffness is present in theWinkler model. Still, the bending deformation comprises under 10% of the gasketdeformation, whereby it can be concluded that the modellingof the Gina gasket witha single spring should be sufficient, since a more sofisticated modelling solely willincrease the rotational stiffness.

10.2.4 Importance of retroaction

In the preceding, it has been shown that the major differencebetween the Winklermodel and the continuum model should be sought for in the axial deformations. Oneof the major principal differences between the models is that the soil springs in the

94

Importance of retroaction

0 5 10 15

−1.5

−1

−0.5

0

0.5

1

1.5x 10

−3

Time, t [s]

Axi

alde

form

atio

n,∆

u[m

] Around y-axis,PitchAroundz-axis,Yaw

(a) Winkler model

0 5 10 15

−1

−0.5

0

0.5

1

1.5

x 10−4

Time, t [s]

Axi

alde

form

atio

n,∆

u[m

] Around y-axis,PitchAroundz-axis,Yaw

(b) Continuum model

Figure 10.11:Axial deformation due to rotations in the gasket centre.

Winkler model are decoupled,i.e. . that the only connection between two adjacentsoil springs is the tunnel.

The two models are sketched in Figure 10.12, where it can be seen that the only thingwhich couples two adjacent springs is the shear stiffness ofthe tunnel. This opposesto the continuum model, where the soil elements around the tunnel are coupled, mak-ing retroaction possible,i.e. waves can be reflected from the tunnel back to the soilelements, and the soil-structure interaction is dependenton the adjacent strain state.

Figure 10.12:The Winkler model (topmost) and the continuum model. The gaskets arecoloured red, the tunnel black, and the soil grey. The blue dotted lines indicatethat no retroaction is possible in the Winkler model.

95


This difference between the two model is not easy to eradicate. It should be clearthat the continuum model in this respect is the better estimate of the physical prob-lem. To incorporate retroaction in the Winkler model, a soilelement with appropriateparameters could be connected to the tunnel, as it is sketched in Figure 10.13. Thesoil element should model the soil behaviour, and its parameters could include bothvarious stiffness’s, damping and mass. This should be able to provide a better modelrepresentation. However, the determination of the soil element parameters is far fromtrivial, and it has not been investigated further in the present thesis.

Figure 10.13:The Winkler model of Figure 10.12 fitted with a soil element to accountfor retroaction. The input displacements should still be applied to the lowersprings. Only vertical soil springs are depicted.

Instead, the consequence of removing the retroaction possibility in the continuummodel has been analysed. This should illustrate the sense ofretroaction modelling. Inthe preceding, the deformations analysed are in the gasketsand therefore, the retroac-tion possibility has been removed over the gaskets. Since the tunnel elements arevery stiff compared to the gaskets, it is also deemed that theeffect of removing theretroaction possibility will be greatest there.

Separation planes

To take away the the possibility of retroaction, separationsoil planes have been imple-mented in the continuum model in the gasket planes, as it is sketched in Figure 10.14.The elements in the separation planes are modelled without stiffness and mass, thus re-moving all ability to transfer any force or displacement. Hence, the continuum modelimitates the behaviour of the Winkler model, as it is sketched in Figure 10.12. How-ever, the element surfaces outlined with a green line in Figure 10.14 will be modelledwith a “free” surface, thus making the soil too flexible. Thisshould be rememberedwhen interpreting the results of the analysis.

Figure 10.14: Insertion of a separation plane into the continuum model of Figure 10.12.Theblue soil elements have no stiffness or mass, and the green line indicates ele-ments with a “free” surface.

96

Spring planes

The deformed domain is depicted in Figure 10.15, where the separation layers clearlycan be observed.

Figure 10.15:Deformed domain for an analysis with separation planes. An animated versionof the figure is provided on the enclosed DVD.

The deformation for the chosen gasket corner (Figure 10.1) is plotted in Figure 10.16.It can be seen that the deformations have been dramatically increased when comparedto Figure 10.6. The total deformation has been increased approximately 30 times, andeven more for the vertical deformation. Due to the “free” surfaces in the gasket crosssections, the gasket deformation now is more than twice the deformation calculated inthe Winkler model.

0 5 10 15

−0.1

−0.05

0

0.05

Time, t [s]

Def

orm

atio

n,∆

u[m

]


Figure 10.16:Deformation time series for the gasket corner in Figure 10.1. The continuummodel with separation planes. The maximum opening and compression ofthecorner is68.7mm and−68.2mm, respectively.

Spring planes

Apparently, the connection of the soil over the gaskets is ofvery high importance. Tofurther investigate the phenomenon, a different approach for modelling of the separa-tion planes is made. In Figure 10.17 on the following page, four different modellingsare sketched. Subfigure (a) shows the modelling without separation planes and (d)illustrates the already performed analysis with no stiffness in the separation planes.

To analyse whether it is the normal stiffness or the shear stiffness which is of im-portance in the separation layers, (b) can be used as modelling, disregarding only theshear stiffness. Furthermore, to investigate the importance of the normal stiffness inthe tunnel-axial direction, the horizontal springs are omitted in (c).

97


(a) Continuum (b) Normal springs (c) Vertical spring (d) No stiffness

Figure 10.17:Different ways of modelling the separation plane elements. The arrows outsidethe elements illustrate the stresses which can be applied to the elements.

The analysis of the two new separation plane models yields the results plotted in Figure 10.18.In Figure 10.18b, the absence of shear stiffness in the separation plane has increasedthe total opening from2mm to 14mm. The vertical deformation has increased muchmore, since very little stiffness in this direction is in themodel. As the axial spring inthe separation plane is removed, the deformation raises to the level of the separationplane without stiffness. This can be seen in Figure 10.18b, which in every aspect isnearly identical to Figure 10.16

0 5 10 15−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Time, t [s]

Def

orm

atio

n,∆

u[m

]


(a) Normal spring separation planes, cf. Figure 10.17b. The maximum opening and compression of thecorner is14.4mm and−4.5mm, respectively.

0 5 10 15

−0.1

−0.05

0

0.05

Time, t [s]

Def

orm

atio

n,∆

u[m

]


(b) Vertical spring separation planes, cf. Figure 10.17c. The maximum opening and compression of thecorner is68.7mm and−68.2mm, respectively.

Figure 10.18:Deformation time series for the gasket corner in Figure 10.1.

It is clearly seen that the more important property for the separation layer is the stiff-ness in the tunnel-axial (x-axial) direction. If the stiffness disappears, the model’sability to take retroaction into account disappear fully, and this increases the gasketdeformation dramatically.

On the basis of the performed analysis, it is deemed that the presence ofx-axial stiff-

98

Modelling without gaskets

ness in the soil next to the gaskets is the single most important reason for the majordifference between the results from the Winkler and the continuum model.

But still, the results with the modellings of Figure 10.17 are not very close to theresults obtain with the Winkler model. The reason for this isthat the introductionof a “separation layer” also entails introduction of a “free” surface, as it is discussedin the above and depicted in Figure 10.14. Nevertheless the performed analysis verysignificantly spots the importance of incorporating the possibility of retroaction intothe calculation models.

10.2.5 Modelling without gaskets

Apparently, the presented Winkler model does not perform well with the immersedtunnel of the present thesis, since it fails to model the single most important factor inthe soil-structure interaction: the retroaction in the tunnel-axial direction.

However, Winkler models are widely applied. The major flaw inthe present applica-tion is deemed to be the segmentation of the immersed tunnel;i.e. the cross sectionis not uniform, since the very flexible gaskets separate the tunnel elements. Thus, theminor errors are lumped in the gaskets, making the result unusable.

To test the Winkler model’s performance for a uniform cross section, the gaskets inthe models have been removed and replaced by concrete finite elements, thus makingthe immersed tunnel consist of only one single tunnel element. The deformation inthe point shown in Figure 10.1 is still used for the calculation, and the time series areshown in Figure 10.19 on the following page.

It can be seen that the deformation time series are very alikefor the Winkler and thecontinuum model if the gaskets are removed. Thus, the Winkler model is a goodmodel for dynamic analysis of underground elongated structures, if the cross sectionis uniform. The gaskets of the immersed tunnel makes the model overly conservativein the gasket deformation calculations.

The similarity of the plots in Figure 10.19 also verifies the model coding of both theWinkler model and the continuum model.

Even though the Winkler model in this chapter has been shown to yield results verydifferent from the continuum model with respect to the deformation of the gaskets, themodel will still be used together with the continuum model inthe following. This isdone in order to examine, whether the much simpler Winkler model could be workablefor parameter studies.

99


0 5 10 15

−4

−2

0

2

4

6x 10

−5

Time, t [s]

Def

orm

atio

n,∆

u[m

]TotalAxial, xTransverse,yVertical, z

(a) Winkler model

0 5 10 15

−4

−2

0

2

4

6x 10

−5

Time, t [s]

Def

orm

atio

n,∆

u[m

]


(b) Continuum model

Figure 10.19:Deformation time series for the point in Figure 10.1. The gaskets are replacedby concrete finite elements.

10.3 Winkler soil springs

The determination of the soil springs in the Winkler model isdescribed in Section 8.2.1.The soil spring stiffness’s are determined on the basis of anelastic model, thus mak-ing the results comparable to the continuum model. If a different material model wasapplied (e.g. to account for non-linear behaviour), if the stratificationof the projectsite was different, or simply if the material stiffness parameter should be determinedto something different, the soil spring stiffness’s could vary.

To account for the influence of the magnitude of the soil spring stiffness’s, a sensi-tivity analysis have been performed. The soil spring stiffness’s have been scaled withdifferent common factors and the resultant maximal gasket deformation has been cal-culated. For stiffness scales from 0.1 to 10, the result is plotted in Figure 10.20 on thenext page.

It can be seen that for moderate variations of the soil springstiffness’s,±50%, thegasket deformation only varies with around±15%. Thus, the soil spring stiffness’scan be determined with some scatter without comprising the accuracy of the Winklermodel. However, if changes in the soil spring stiffness should be due to changes in thegeneral soil stiffness, and hence changes in the wave velocity in the soil, the free-fieldsoil response will change significantly cf. Section 11.3 on page 106.

100

Prestress in tunnel

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

Soil spring stiffness scale

Nor

mal

ized

max

.ga

sket

defo

rmat

ion

Figure 10.20:The frequency response function for other depths of layerC.

In Figure 10.20 it can further be observed that the gasket deformation decreases monotonouswith increasing soil spring stiffness. This was also to be expected, since an increasingsoil spring stiffness will make the tunnel follow the prescribed displacements moreclosely, and the deformation will decrease towards a minimum specified by the appar-ent velocity.

10.4 Prestress in tunnel

As described in Section 5.6.1, the tunnel elements are prestressed during the instal-lation phase. The purpose of the prestress is to compress theGina gaskets, but theinduced stresses in the system could also affect the stiffness of the tunnel elements, asit is known from an ordinary beam-column problem. An axial compressive stress willreduce the bending stiffness against cross axial load.

The prestressing force cannot be neglected offhand, since the total force is of magni-tude 48 000kN, cf. (5.13). However, in the present chapter it is repeatedly shown thatthe axial compression of the immersed tunnel is far more significant than bending.Thus, it is deemed that prestress in the tunnel will have negligible influence on thetunnel damage. To support is assumption, a sensitivity analysis has been carried out.

The bending stiffness have been scaled by a varying factor, by scaling of the crosssectional second moments of area,Iy and Iz . In Figure 10.21 on the next page themaximum deformation has been plotted for the reference gasket corner.

It can be seen that the gasket deformation is very insensitive to variations in the bend-ing stiffness of the tunnel elements – only a slight decreasein the deformation occursfrom increasing bending stiffness. This can be explained bytwo characteristics of thesystem, which has also been verified in the above: 1) the axialdeformation is of muchgreater importance than bending of the tunnel, and 2) since the gasket stiffness is muchless than the tunnel stiffness – even if the the bending stiffness of the tunnel is scaled –the free-field soil displacements are lumped in the gaskets.

101


0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.995

1

1.005

1.01

1.015

Scaling of second moments of area

Nor

mal

ized

max

.de

form

atio

n

Figure 10.21:Sensitivity of the tunnel element bending stiffness. Normalized deformation inthe gasket corner in Figure 10.1.

102

CH

AP

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11SOIL PARAMETERS AND STRATIFICATION

In Chapter 5 the soil stratification and the soil parameters for the project are presented,and in Section 6.5 the response at the level of the tunnel is calculated by means of thedomain transformation method. In this chapter the sensitivity of the soil response withrespect to changes in the stratification and in the soil parameters is analysed.

The purpose of the sensitivity analyses are to indicate, howmuch time and moneyshould be spend in the determination of the stratification and soil parameters. If thesoil response is insensitive to changes ine.g. the layer thickness, only minimal soilinvestigation is needed. On the other hand, if great sensitivity is present, it may befavourable to invest time and money in detailed soil investigations.

11.1 Method of analysis

In the following analyses of this chapter, the sensitivity parameter is chosen as themaximum absolute soil response from the time series at the level of the tunnel, cal-culated with the domain transformation method. For the reference response calcu-lated with the parameters given in Chapter 5, this value is175mm cf. Figure 6.12 onpage 50. The data of the plots in this chapter have been normalized with this value.For comparison, the maximum absolute displacement of the earthquake time series is95.5mm cf. Figure 5.7.

The choice of this sensitivity parameter is made because it is well correlated to the de-formation in the gasket, which is the final design parameter cf. Section 5.7 on page 38.Therefore, there is no need to perform excessive calculations to determine the finalgasket deformation for these sensitivity analyses. The correlation has been verified inSection 12.3, where it has been shown that because the modelsare linear, the gasketdeformations are directly proportional to the free-field soil response.

103

Chapter 11. Soil parameters and stratification

Even though the maximum absolute displacement may not be a perfect representationof a time series – since all other soil response data is not used for the sensitivityanalyses – it can be seen by comparison ofe.g.Figure 6.12 on page 50 and Figure 10.5on page 90 that the maximum gasket deformation occurs as the immediate reaction tothe maximum soil response. Thus, the maximum absolute displacement may withgood accuracy be used as indirect indication of the tunnel damage.

If the layer interfaces are not horizontal and the stratification similar at the entireproject site, incoherence may arise cf. 4.11c on page 26. This type of incoherencehas not been analysed in the present thesis, as it is stated inSection 4.6. Since theanalyses of the wave propagation is purely one-dimensionalit would not make muchsense to make detailed studies with these methods, as the wave propagation will bethree-dimensional for varying stratification.

The exact sensitivities calculated will vary for differentearthquakes with varying fre-quency spectra. However, the following analyses should nevertheless provide clearindications of the important factors.

11.2 Impact of layer depths

The stratification may not be very well determined in the deeper subsoil, since deepborings are very expensive.

In Figure 5.1 on page 28 the stratification for the longitudinal section is depicted. Itcan be seen that the layer thicknesses vary over the tunnel. In Section 5.4 it is shownhow mean thicknesses have been chosen as the design basis forthe entire domain.Thus, the geographic variations are not included in this thesis. In this section, theconsequences of different thicknesses for the soil layers have been analysed.

11.2.1 Thickness of layer A

The maximum displacement at the level of the tunnel as function of the thickness oflayerA is plotted in Figure 11.1. Only minor sensitivity is present, since it can be seenthat if e.g.the layer vanishes, the displacement amplitude increases by around1%.

0 1 2 3 4 5 6 7 8 9 100.995

1

1.005

1.01

Layer thickness, layerA [m]

Nor

mal

ized

max

.di

spla

cem

enta

ttun

nel

Figure 11.1: Influence of the thickness of layerA.

104

A

Thickness of layer B

11.2.2 Thickness of layer B

The sensitivity of the displacements with regard to the thickness of layerB is plottedin Figure 11.2. The deformation may be increased with up to7%, should the layervanish.

0 2 4 6 8 10 12 14 16 18 20

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

Layer thickness, layerB [m]

Nor

mal

ized

max

.di

spla

cem

enta

ttun

nel

Figure 11.2: Influence of the thickness of layerB.

11.2.3 Distance to bedrock

In (COWI 2007) the thickness of layerC is estimated to100−150m as mentioned inSection 5.2. Since the thickness of the layer,i.e. the distance to the bedrock, is not welldetermined, the consequences of different thicknesses forlayerC has been analysed.The result is plotted in Figure 11.3.

0 50 100 150 200 250 300

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Layer thickness, layerC [m]

Nor

mal

ized

max

.di

spla

cem

enta

ttun

nel

Figure 11.3: Influence of the thickness of layerC.

It can be seen that rather large sensitivity is present. Should the lower value of theestimated interval,100m, be the correct value of the layer thickness, the soil displace-ment is magnified with20%. Even a magnification of approximately30% is possibleif the correct value should be80m.

In general, it can be concluded that the stratification is of rather high importance forthe calculated displacements at the level of the tunnel. In areal-world design pro-cess, the exact stratification will never be known. Therefore, care should be takenwhen determining the stratification, and for a final design sensitivity analyses like the

105

B

C

C


above should be performed, and the design stratification chosen from the maximumdisplacement. Also, if the layer interfaces diverges significantly from being horizontal,this must be taken into consideration.

11.3 Wave velocity

The wave velocity depends on the shear stiffness and the density of the soil, as givenby (A.4) on page 130. Therefore, the result for a sensitivityanalysis of the wavevelocity also applies for the soil shear stiffness and the soil density.

The sensitivities for varying wave velocities are plotted in Figure 11.4. Like it isdiscussed for the layer thicknesses, it can be seen that the design wave velocity cannot be chosen on the safe side as an upper or lower value. Especially for layer C thewave velocity is significant, and magnifications of the displacements may be as highas30% for a 30% increase in wave speed. For a final design, the expected intervalof the wave velocities should be determined, and sensitivity analyses performed todetermine the maximum displacement.

100 150 200 250 300 350 400 450 500

0.9997

0.9998

0.9999

1

1.0001

1.0002

S-wave velocity for layerA[

ms

]

Nor

mal

ized

max

.di

spla

cem

enta

ttun

nel

(a) LayerA

200 250 300 350 400 450 500 550 6000.99

1

1.01

1.02

1.03

1.04

1.05

1.06

Nor

mal

ized

max

.di

spla

cem

enta

ttun

nel

S-wave velocity for layerB[

ms

]

(b) LayerB

Figure 11.4: Influence of the wave velocity of the soil layers.

106

C

Damping

300 400 500 600 700 800 900 1000

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

ylab

el

S-wave velocity for layerC[

ms

]

(c) LayerC

Figure 11.4: Influence of the wave velocity of the soil layers (Continued).

11.4 Damping

The application of damping in the thesis is outlined in Section 5.3. Here, the conse-quences are analysed.

11.4.1 Sensitivity

The sensitivity of the soil response with respect to the lossfactor of the soil layers isplotted in Figure 11.5. Due to its greater thickness, layerC is the only layer showinga significant sensitivity. In general, it can be observed that more damping generatesless soil response, as it would also be intuitively expected. However, it may be seenthat the sensitivity of the the soil response is very little,since only up to3% of thedisplacement fades for moderately high damping of soilC, η= 0.1.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.97

0.98

0.99

1

1.01

LayerALayerBLayerC

Loss factor[-]

Nor

mal

ized

max

.di

spla

cem

enta

ttun

nel

Figure 11.5: Influence of the damping loss factorη.

The effect of damping can also easily be seen in the frequencyresponse function whichis plotted in Figure 11.6. Especially the response in the eigenmodes is significantlydamped.

107

C

C

A

B

C


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

0

101

102

103

η= 0η= 0.01η= 0.02η= 0.03η= 0.04

Frequency,f Hz

Fre

quen

cyre

spon

se,

H[-

]

Figure 11.6: Frequency response function for the stratum at the level of the tunnel.

11.4.2 Hysteretic damping and viscous damping

Both hysteretic damping and viscous damping have been integrated in the domaintransformation method, and two time series at the level of the tunnel have been cal-culated in the frequency domain and plotted in Figure 11.7 onthe next page. It canbe seen that very little difference exists between the two damping mechanisms. Thiswas also the conclusion in Section 9.6.1, where the time series calculated with hys-teretic damping has been applied to the continuum model, whose solution in the timedomain entails application of viscous damping. In fact, it can be seen that the plot ofFigure 11.7 is very similar to Figure 9.15 on page 80. This indicates that the coding ofdamping in the continuum model and the domain transformation method yields verysimilar results, thus verifying the coding of both models.

108

Hysteretic damping and viscous damping

0 5 10 15

−0.15

−0.1

−0.05

0

0.05

Hysteretic dampingViscous damping

Time, t [s]

Dis

plac

emen

t,u[m

]

(a) Time series for the first15s

5 5.5 6 6.5 7 7.5 8 8.50

0.02

0.04

0.06

0.08

0.1

Hysteretic dampingViscous damping

Time, t [s]

Dis

plac

emen

t,u[m

]

(b) Close-up

Figure 11.7: The soil response at the level of the tunnel. Viscous damping has been calibratedfor the hysteretic damping at a frequency of1.09Hz.

109

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12THE EARTHQUAKE

In this chapter sensitivity analyses are performed for the apparent velocity, the direc-tion and the displacement amplitude of the earthquake.

The damage measure is for the continuum and the Winkler models chosen to the de-formation of the gasket corner depicted in Figure 10.1. Thisis chosen instead of themaximal deformation at all gasket corners, since a comparison is deemed to be moreappropriate and understandable when a specific node is used.For the closed formsolution, only a single gasket deformation value exists.

12.1 Apparent velocity

Firstly, the sensitivity of the damage to the tunnel with respect to the apparent velocityis analysed. Different values of the apparent velocity havebeen used for simulations inthe continuum model, and the maximum opening and compression of the gasket cornershown in Figure 10.1 has been plotted in Figure 12.1a. Even though the Winklermodel and the closed form solution in Chapter 10 has been shown to be unreliablefor the immersed tunnel, calculations with the Winkler model and the closed formsolution have also been made, due to the simple formulations. The results are plottedin Figure 12.1b. The ease of making parameter studies with the Winkler model andthe closed form solution, which is coded by hand inMATLAB , is the reason for theadditional data.

Even though the models in Figure 12.1 do not yield the same results, a similar trend isclearly shown. The similarity between the models is furtheroutlined in Figure 12.2 onpage 113, where the results of Figure 12.1 are plotted for thethree models together.

It can be seen that changes in apparent velocity of±500 ms

yield changes in the gasketdeformation of around∓20%. Since the greater damage occurs for lower values of the

111

Chapter 12. The earthquake

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000−3

−2

−1

0

1

2

3x 10

−3

Apparent velocity[

ms

]

Def

orm

atio

n,∆

u[m

]

OpeningCompression

(a) Continuum model

500 1000 1500 2000 2500 3000

−0.05

0

0.05

0.1

0.15

Apparent velocity[

ms

]

Def

orm

atio

n,∆

u[m

]

Winkler - OpeningWinkler - CompressionClosed form solution

(b) Winkler model and closed form solution

Figure 12.1: Maximum deformation in gasket corner shown in Figure 10.1.

apparent velocity, the lowest reasonable value should be used for a final design.

It does seem very reasonable that the general trend is increasing apparent velocityfor decreasing damage of the tunnel, since the incoherence stems from the apparentvelocity, and the incoherence is closely related to the damage of the tunnel, as it is dis-cussed in Section 4.6. However, in a dynamic analyses it could normally be expectedthat certain eigenmodes of the system would be excitated from certain apparent ve-locities. Thus, a decreasing tendency could be expected, but some local amplificationwould not be surprising to observe.

In Figure 12.2 however, the deformation is decreasing monotonously with increasingapparent velocity. The reason for this must be found in the discussion of Section 4.4;that the behaviour of an underground structures is not dominated by the inertia of thestructure. Hence, the most important seismic parameter forunderground structures isnot the soil acceleration but the soil displacement, and theeigenmodes of the immersedtunnel become of minor importance.

112

Direction

500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5

Apparent velocity[

ms

]

Nor

mal

ized

defo

rmat

ion Continuum model

Winkler modelClosed form solution

(a) Gasket opening

500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5

Apparent velocity[

ms

]

Nor

mal

ized

defo

rmat

ion Continuum model

Winkler modelClosed form solution

(b) Gasket compression

Figure 12.2: Results from Figure 12.1 with deformation normalized for an apparent velocityof 1500 m

s .

12.2 Direction

Similar to the previous section simulations have been performed, now with varyingangle of direction. Since the system is double symmetric, only angles fromθ = 0◦ toθ = 90◦ are used for the analysis. The definition of the angle is provided in Figure 5.9on page 35.

The gasket opening and compression as function of the angle of propagation are de-picted in Figure 12.3 for the continuum model, the Winkler model, and the closed formsolution. It is clearly seen that the maximal gasket deformation occur with a directionangle of approximately45◦. Thus, as it was deemed in Section 5.5.3, the critical angleof approach is oblique, where both axial particle motion, cross axial particle motionand an appropriate incoherence are present.

It should be noted that the calculated deformation is∆u = 0 for θ = 0. This is becauseno incoherence is present when the wave propagates perpendicular to the tunnel.

As in the previous section, the trend in Figure 12.3 is very similar for all three models.This is more evidently seen in Figure 12.4, where the deformations has been normal-ized. Thus, it is indicated that both the Winkler model and the closed form solution,even though the absolute values of the calculated deformations are highly conserva-tive, advantageous could be used for some parameter studies.

113


0 10 20 30 40 50 60 70 80 90−3

−2

−1

0

1

2

3x 10

−3

Direction angle[◦]

Def

orm

atio

n,∆

u[m

]

OpeningCompression

(a) Continuum model

0 10 20 30 40 50 60 70 80 90−0.04

−0.02

0

0.02

0.04

0.06


Def

orm

atio

n,∆

u[m

]

Winkler - OpeningWinkler - CompressionClosed form solution

(b) Winkler model


114

Direction

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1


Nor

mal

ized

defo

rmat

ion

Continuum modelWinkler modelClosed form solution

(a) Gasket opening

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1


Nor

mal

ized

defo

rmat

ion

Continuum modelWinkler modelClosed form solution


Figure 12.4: Results from Figure 12.3 with deformation normalized for an angle of45◦.

115


12.3 Earthquake amplitude

The model is fully linear, and therefore should the gasket deformations – together withall other displacements and their derivatives – be directlyproportional to a scaling ofthe input displacements. This has been verified through calculations in the Winklerand the continuum model; the closed form solution is omittedsince the proportionalityis very obvious seen from (7.1) on page 53. The displacement time series has beenscaled with a varying factor, and the results are plotted in Figure 12.5.

0.4 0.6 0.8 1 1.2 1.4 1.60.4

0.6

0.8

1

1.2

1.4

1.6

Amplitude magnification factor[-]

Nor

mal

ized

defo

rmat

ion

ContinuumWinkler

(a) Gasket opening

0.4 0.6 0.8 1 1.2 1.4 1.60.4

0.6

0.8

1

1.2

1.4

1.6

Amplitude magnification factor[-]

Nor

mal

ized

defo

rmat

ion

ContinuumWinkler


Figure 12.5: Normalized maximum deformation in gasket corner shown in Figure 10.1.

It can be seen that a scaling of the input displacement time series yields the very samescaling of the gasket deformations, thus verifying the linearity of the models. In thereal world tunnel, this will only apply if the deformation state in the soil and structureis in the range where linear behaviour can be approximated.

116

CH

AP

TE

R

13M ODELLING OF GASKET JOINTS

As it is discussed in Section 5.6, the modelling of the Gina gaskets is not trivial. Alinear approximation has been made, all though the behaviour is highly non-linear,both in the longitudinal and in the transverse and vertical direction.

As in Chapter 12, the continuum model and the Winkler model are used for the sensi-tivity analyses. However, since the gasket behaviour is notincluded in the closed formsolution, this is omitted.

13.1 Longitudinal stiffness

The axial (longitudinal) stiffness of the Gina gaskets is highly non-linear, as presentedin Section 5.6.1. A linearization has been made based on the gasket compressionduring the installation of the tunnel. As the gasket is compressed or opened, the actualstiffness will change rapidly. To give an indication of the impact of the linearization,a sensitivity analysis has been carried out.

The longitudinal gasket stiffness has been scaled with a factor, and in Figure 13.1 onthe following page the damage to the reference gasket corneris plotted as function ofthe factor, for both the Winkler model and the continuum model.

As in Chapter 12 it is seen that the trend seems quite similar for the Winkler modeland the continuum model. In Figure 13.2 this is more evidently seen as the normalizedgasket deformation is plotted. The general trend is a slightdecrease in the gasketdeformation as the axial gasket stiffness is increased, as it would also be expectedfrom a general static point of view. However, the change is very little, especiallyfor the continuum model which is regarded the better one. This indicates that themodelling of the axial gasket stiffness shouldnot be given very deep considerationwhen modelling the structure, since a variation in the stiffness does not affect the

117

Chapter 13. Modelling of gasket joints

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4−3

−2

−1

0

1

2

3x 10

−3

Factor on axial stiffness[-]

Def

orm

atio

n,∆

u[m

]

OpeningCompression

(a) Continuum model

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03


Def

orm

atio

n,∆

u[m

]

OpeningCompression

(b) Winkler model


gasket deformation significantly.

The results may be surprising, since it is documented in Chapter 10 that the axial defor-mation is very highly affective on the gasket deformation. The reason that variationsof the axial gasket stiffness has very little effect on the gasket deformation is deemedto be the great difference between the stiffness of the tunnel elements and the axialgasket stiffness. The axial gasket stiffness make up approximately 3.98·106

40·109 = 0.1‰ ofthe tunnel element stiffness. Since the tunnel is surrounded by soil, the overall dis-placements are governed by the soil motion and thus, it does not make very muchdifference if the gasket stiffness comprises0.10‰ or 0.15‰ of the tunnel stiffness –the majority of the deformation is still lumped in the gasket.

118

Shear stiffness

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.95

1

1.05


Nor

mal

ized

defo

rmat

ion Continuum

Winkler

(a) Gasket opening

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.95

1

1.05


Nor

mal

ized

defo

rmat

ion Continuum

Winkler


Figure 13.2: Results from Figure 13.1 with deformation normalized for the initial lineariza-tion of the gasket stiffness.

13.2 Shear stiffness

The choice of an appropriate shear stiffness for the gasket is discussed in Section 5.6.2.In this thesis, it has been chosen to use a shear stiffness of the joint corresponding tothe shear stiffness of the rubber gasket itself, disregarding the shear keys of concrete.

As a sensitivity analysis, the gasket shear stiffness have been set to both zero andequivalent to the shear stiffness of the concrete tunnel elements, in two different cal-culations. The shear stiffness equivalent to concrete is used to model the shear keysin action as an upper value of the stiffness. For the continuum model this correspondsto shear modules ofG = 0Pa andG = 17.4 ·109 Pa, respectively. For comparison, thereference shear modulus of the rubber gasket profile isG = 1.3 · 106 Pa. The resultsfor the Winkler model and for the continuum model are depicted in Figure 13.3 andFigure 13.4.

It is clearly seen that the magnitude of the gasket shear stiffness does not influence sig-nificantly on the gasket deformation. Thus, as for the axial gasket stiffness, the crossaxial gasket stiffness should not be given significant considerations when modellingthe immersed tunnel for a final design. The reason for this is primarily deemed to bethe definition of damage in the present thesis, since transverse deformation does notcontribute significantly to the total gasket deformation. In Figure 13.5 the maximumtransverse deformation is plotted. It can be seen that the introduction of a shear stiff-

119

Chapter 13. Modelling of gasket joints

−3

−2

−1

0

1

2

3x 10

−3

No shear stiffness Rubber shear stiffness Concrete shear stiffness

Def

orm

atio

n,∆

u[m

]

OpeningCompression

(a) Continuum model

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03


Def

orm

atio

n,∆

u[m

]

OpeningCompression

(b) Winkler model


ness like concretedoesdecrease the transverse gasket deformation significantly,from1.1mm to 0.06mm, but this does not influence significantly on the final deformation.

120

Shear stiffness

0.9992

0.9994

0.9996

0.9998

1

1.0002

1.0004

1.0006


Nor

mal

ized

defo

rmat

ion

ContinuumWinkler

(a) Gasket opening

0.9985

0.999

0.9995

1

1.0005


Nor

mal

ized

defo

rmat

ion

ContinuumWinkler


Figure 13.4: Results from Figure 13.3 with deformation normalized for the rubber shear stiff-ness.

0

0.2

0.4

0.6

0.8

1

1.2x 10

−3


Def

orm

atio

n,∆

u[m

]

Figure 13.5: Maximum transverse deformation in gasket corner shown in Figure 10.1, for thecontinuum model.

121

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126

www.trelleborg.com/bakker/files/pdf_eng/Gina_eng.pdf

www.trelleborg.com/bakker/files/pdf_eng/Gina_eng.pdf

APPENDICES

127

AP

PE

ND

IX

ADERIVATION OF THE DOMAIN

TRANSFORMATION METHOD

In this section the domain transformation method is derived, based upon (Andersen2006). The bases for the method are outlined in Chapter 6, andthe coordinates andlayer indices used refer to Figure 6.2 on page 44.

A.1 Constitutive model

The material model is in Section 6.1 given as homogeneous, isotropic and linear vis-coelastic. The general constitutive model for a layer of such material is given by

σj

i k=λ j ǫl lδi k +2µ j ǫi k , ǫ

j

i k=

1

2

(

∂Uj

i

∂xk+∂U

j

k

∂xi

)

(A.1)

(Byskov 2002, p96), where thehat ( ) signifies the formulation in the frequency do-main. For hysteretic damping to be applied, the Lamé constantsλ andµ are expressedin terms of Young’s modulusE , Poisson’s ratioν, the circular frequencyω and theloss factorη as

λ=λ′·(

1+ isign(ω)η)

, λ′=

νE

(1+ν)(1−2ν)

µ=µ′·(

1+ isign(ω)η)

, µ′=

E

2(1+ν)

(A.2)

(Andersen 2007)

129

Appendix A. Derivation of the domain transformation method

A.2 Governing equation

The layers( j = 1. . . J) are one by one regarded as linear elastic, isotropic and homo-geneous media. The equations of motion for such media are forthe general three-dimensional case theNavier equations

(λ+µ)∂2u j

∂xi∂x j+µ

∂2ui

∂x j∂x j+ρbi = ρ

∂2ui

∂t 2(A.3)

where the conventions of index notation applies,ui is the displacement in the directionof the coordinatexi , ρ is the mass density,bi is the body forces per unit mass andt isthe time coordinate.

In (A.3) nearly all terms reduces to zero when the simplified problem is observed.Since a vertically propagating SH-wave is considered, no vertical movement occurs,i.e. u

j3 = 0, with j and3 indicating thej th layer and thex3-coordinate, respectively.

Furthermore, the system of coordinates is oriented such that the wave motion is in thex1-direction, providingu

j2 = 0.

The wave propagates in thex3-direction. Thus, for a givenx3-coordinate the samedisplacement occurs for allx1 andx2, i.e.∂u

j

i/∂x1 = ∂u

j

i/∂x2 = 0.

If these simplifications for the SH-wave are incorporated into (A.3) and body forcesare disregarded, the following one-dimensional wave equation, which governs thewave propagation, is found.

∂2u j (z, t )

∂z2=

1

(cj

S)2

∂2u j (z, t )

∂t 2, z = x3, u j (z, t ) = u

j1(x3, t ), c

j

S=

√

µ j

ρ j(A.4)

where thez-coordinate is introduced together with the shear wave velocity cS .

A.3 Transformation into frequency domain

The governing equation (A.4) is formulated in the time domain (indicated by the pres-ence of the time coordinate). The relation between the time and frequency domainrepresentation of the displacement is provided by the inverse Fourier transformation

u j (z, t ) =1

2π

∞∫

−∞

U j (z,ω)eiωt dω (A.5)

The integral in (A.5) can be discretised to a complex Fourierseries; a sum ofN discreteharmonic waves with the frequencyωn as:

u j (z, t ) ≈N∑

n=1

Uj

n(z)eiωn t (A.6)

130

Relation between the layers

Each term of (A.6),u jn(z, t ) = U

jn(z)eiωn t , can be inserted into (A.4). The second-

order derivatives of the term are

∂2ujn(z, t )

∂z2=

∂2Uj

n(z)

∂z2eiωn t

∂2ujn(z, t )

∂t 2= (iωn)2U

jn(z)eiωn t

=−ω2nU

jn(z)eiωn t

(A.7)

and applied in (A.4) this yields

∂2Uj

n(z)

∂z2eiωn t

=−1

(cj

S)2ω2

nUj

n(z)eiωn t

∂2Uj

n(z)

∂z2=−

1

(cj

S)2ω2

nUj

n(z) =−(kjn)2U

jn(z) (A.8)

where the wave numberkjn = ωn/c

j

Sis introduced. (A.8) represents the equation of

motion in the frequency domain for thenth frequency of thej th layer.

A local z j -axis is applied for every layer, as shown in Figure 6.2 on page 44. Theheight of each layer is denotedh j , and (A.8) is a constant-coefficient differential equa-tion with the solution

Uj

n(z j ) = Bjn eik

jn z j

+Cjn e− ik

jn (z j −h j ) (A.9)

whereBjn andC

jn are integration constants.

Now the stresses can be found from the displacements throughthe constitutive re-lation (A.1) on page 129. As indicated in Section A.2, only the partial derivative∂U

j1 /∂x3 is different from zero, which simplifies (A.1) to

σj13 = σ

j31 =µ j

∂Uj

1

∂x3(A.10)

By utilization of (A.9) andx3 = z, and by introducing the stress amplitudePjn(z j ) =

σj13, (A.10) ends up to

Pjn(z j ) = ik

jnµ

j(

Bjn eik

jn z j

−Cjn eik

jn (z j −h j )

)

(A.11)

A.4 Relation between the layers

Two auxiliary matricesS jn andA

jn which gathers the strain and stresses, are now intro-

duced as

Sjn(z j ) =

[

Uj

n(z j )

Pjn(z j )

]

= Ajn(z j )

[

Bjn

Cjn

]

(A.12)

Ajn is defined according to (A.9) and (A.11) as

Ajn(z j ) =

[

eikjn z j

e− ikjn (z j −h j )

ikjnµ

j eikjn z j

− ikjnµ

j e− ikjn (z j −h j )

]

(A.13)

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Appendix A. Derivation of the domain transformation method

Now, the relation between the strain and stressesSjn at the top and the bottom of a

layer can be found. Superscript 0 and 1 signifies the top and bottom of the layer,respectively,i.e. the requested relation is betweenS

j 0n = S

jn(z j = 0) andS

j 1n = S

jn(z j =

h j ). Through evaluation of (A.12) the relation is found as

{

Ajn(z j

= 0)}−1

Sjn(z j

= 0) =

[

Bjn

Cjn

]

=

{

Ajn(z j

= h j )}−1

Sjn(z j

= h j )

⇒ Sj 0n = A

j 0n

{

Aj 1n

}−1S

j 1n (A.14)

It is observed thatS j 0n is found through simple multiplication of matrices ontoS

j 1n , a

very fast operation for a computer, relative to inversion oflarge matrices.

Continuity is now required in the interfaces between the layers. In other words, itis required that the displacement is identical in the interface and that equilibrium isfulfilled, i.e. S

j 0n = S

j−1,1n . This demand makes it possible to establish a simple relation

between the top of the topmost layerS10n and the bottom of the bottommost layerS

J1n

through reiterated application of (A.14).

S10n =A

10n

{

A11n

}−1S

11n = A

10n

{

A11n

}−1S

20n = A

10n

{

A11n

}−1A

20n

{

A21n

}−1S

21n = . . .

=A10n

{

A11n

}−1A

20n

{

A21n

}−1· · ·A

J0n

{

AJ1n

}−1S

J1n (A.15)

Thetransfer matrixTn is then introduced, which reduces (A.15) to

S10n = T

10n S

J1n , T

10n = A

10n

{

A11n

}−1A

20n

{

A21n

}−1· · ·A

J0n

{

AJ1n

}−1(A.16)

A.5 Boundary conditions

The boundary conditions of the problem are: 1) No shear stress at the top of thetopmost layer,P 10

n = 0, and 2) An earthquake-induced displacement at the bottom ofthe bottommost layer,U J1

n = Un . The boundary conditions are introduced in (A.16)which yields the equation

[

U 10n

0

]

=

[

T 1011 T 10

12

T 1021 T 10

22

][

Un

P J1n

]

(A.17)

Solving (A.17), the displacement at the top of the topmost layer can be found directlyas a function of the earthquake induced displacement.

U 10n =

(

T 1011 −

T 1012 T 10

21

T 1022

)

Un = H 10n Un , H 10

n = T 1011 −

T 1012 T 10

21

T 1022

(A.18)

where the frequency response functionHn has been introduced. This may be com-puted for each harmonic wave component in the earthquake spectra according to (A.6).

132

Solution for inner interfaces

A.6 Solution for inner interfaces

If the deformations and stresses in the interfaces between layers is of interest, they canbe computed easily by a generalization of the transfer matrix Tn . For the top of thej th layerT

j 0n can be defined analogous to (A.16) on the preceding page:

Sj 0n = T

j 0n S

J1n , T

j 0n = A

j 0n

{

Aj 1n

}−1A

j+1,0n

{

Aj+1,1n

}−1· · ·A

J0n

{

AJ1n

}−1(A.19)

The stress amplitude can be found from (A.17):

0 = UnT 1021 +P J1

n T 1022 ⇒ P J1

n =−UnT 10

21

T 1022

(A.20)

Hence, the displacements and stresses gathered inSj 0n can be determined from known

values:

Sj 0n =

[

Uj 0

n

Pj 0n

]

=

[

Tj 0

11 Tj 0

12

Tj 0

21 Tj 0

22

]

[

Un

P J1n

]

(A.21)

where the deformation at the top of thej th layer, analogous to (A.18), can be expressedexplicitly as

Uj 0

n =

(

Tj 0

11 −T

j 012 T 10

21

T 1022

)

Un = Hj 0n Un , H

j 0n = T

j 011 −

Tj 0

12 T 1021

T 1022

(A.22)

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IX

BDERIVATION OF THE FINITE ELEMENT

METHOD FOR WAVE PROPAGATION

In this appendix, the finite element method (FEM) for the frequency response in a stra-tum is derived. The method and its application is outlined inSection 6.3. The methodhas been implemented in aMATLAB program, which is enclosed on the attached DVD.

This application of the FEM is performed in the frequency domain. This differs fromthe application of the FEM in the Winkler model, cf. Chapter 8, which is performedin the time domain.

In the FEM, a system of matrices is established and solved. The damping of the systemis, as stated in Section 6.3, applied as hysteretic damping,which enters in the stiffnessmatrix K. Thus, the equation of motion in the frequency domain is

(

K−ω2M

)

U = F (B.1)

whereK andM are the stiffness and mass matrix, respectively,U andF are the dis-placement and load vector, respectively, andω is the circular frequency.

B.1 Geometry and topology

The general geometry of the problem is sketched in Figure 6.2on page 44. The geo-metry is discretized with finite elements. Each soil layer isdiscretized into a chosennumber of soil elements, as it is depicted in Figure B.1 on thenext page. As it issketched in this figure, only the vertical coordinate,z, determines the node position.The displacement in the nodes are measured on a horizontal coordinate.

The soil elements are connected in nodes, and both the elements and the nodes arenumbered ascending with thez-coordinate. In each layer, the soil parameters and the

135

Appendix B. Derivation of the finite element method for wave propagation

element sizes are identical, which entails that the elementmatrices in each layer alsoare identical.

Layer interface

Layer interface

Surface

Layer 1

Layer 2

Layer 3

......

......

Element 1

Element 2

Element 3

Element 4

Element 5

Element 6

z

Node 1

Node 2Node 3

Node 4Node 5

Node 6Node 7

Node 8

Node 9

Node 10

Node 11

Node 12

Figure B.1: Sketch of geometry and topology for the FEM.

B.2 Element matrices

The element matrices for the soil elements are established on the basis of the chosenshape functions of the elements. As stated in Section 6.3.1,it is chosen to use second-order elements for the analysis. The second-order shape functions require a node in themiddle of the element, as it is shown in Figure B.1. The shape functions are depictedin Figure 6.5 on page 46.

B.2.1 Stiffness matrix

On the basis of the given shape functions, and taken the physics of the problem (shear-ing of a soil column with shear stiffnessG and depthle) into consideration, the stiff-ness matrix for a soil element may be formulated as

Ke =G

3 le

7 −8 1

−8 16 −8

1 −8 7

(B.2)

(Felippa 2004, p32-11), where it should be remembered thatG is modified to modelhysteretic damping.

B.2.2 Mass matrix

The mass matrix may be obtained in a number of ways, since an exact solution tomodel a continuum distributed mass does not exist. The mass matrix has been com-

136

Solution of the system

bined of two ways of treating the mass: alumpedmass matrix and aconsistentmassmatrix.

The lumped mass matrix, (B.3), is a simple diagonal matrix, which models the massas if it where lumped entirely at the nodes of the soil element. The division of themass in portions of1/6, 1/6 and2/3 are obtained according to Simpson’s rule.

Me,lmp =ρ le

6

1 0 0

0 4 0

0 0 1

(B.3)

(Felippa 2004, p31-8)

The consistent mass matrix, (B.5) should in fact be called the stiffness-consistentmass matrix, since it is constructed by using the same shape functions, depicted inFigure 6.5, as used for the construction of the stiffness matrix.

The consistent mass matrix is calculated as

Me,cons =

∫L

0N

T (x)µ(x)N(x)dx (B.4)

(Nielsen 2004, p153), whereN is the shape function matrix,µ is the distributed mass,andL is the element length. For the second-order soil element this yields

Me,cons =ρ le

30

4 2 −1

2 16 2

−1 2 4

(B.5)

(Felippa 2004, p31-8)

The goodness of the mass matrix chosen can be quantified in terms of its ability topreserve linear and angular momentum, and by modelling the dispersive behaviour ofa continuum. It can be shown that the better combination of the lumped and consistentmass matrices is a simple linear combination,

Me =Me,cons +Me,lmp

2(B.6)

(Felippa 2004, p31-5) (Krenk 2001)

The element matrices, the stiffness as well as the mass matrices, are assembled to theglobal system matrices according to the topology.

B.3 Solution of the system

The solution of the system, (B.1), now follows easily. For clarity, a ’dynamic stiffness’is introduced as

Kdyn = K−ω2M (B.7)

The earthquake enters into the system through forced displacements,u J = u, at bedrock.J is the total number of nodes. The force vector,F consists of zeroes. To account forthis, the system is rewritten, similar to what has been showed in Section C.3, where

137

Appendix B. Derivation of the finite element method for wave propagation

further details can be found. The dynamic stiffness matrix are divided into four sub-matrices

Kdyn =

1 · · · (J −1) J

Kdyn,11 Kdyn,12 1···(J

−1

)

Kdyn,21 Kdyn,22 J

(B.8)

and the solution to the system, the unknown displacementsU1, follows as

U1 = Kdyn,11

(

F−Kdyn,12 · u)−1 (B.9)

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IX

CENHANCEMENTS TO FE W INKLER MODEL

The Winkler model, described in Chapter 8, has been coded inMATLAB on the basisof an existing linear FE program for static analysis, by Stærdahl et al. (2007). Theprogram has been altered, so that a dynamic analysis with forced displacements canbe carried out. In this appendix, the necessary enhancements to the existing programare described. What already was part of the program,e.g.the global matrix assemblerand the element stiffness matrix for a beam element, is not described.

Equations of motion

The basic set of equations which has to be solved is the globalequation of motion

Mu(t )+Cu(t )+Ku(t ) = f(t ) (C.1)

whereu signifies the displacement vector for the degrees of freedom, f is the loadvector, M, C and K is the mass, damping and stiffness matrices, respectively,andwhere a dot( ) signifies differentiation with respect to time.

C.1 Damping and mass matrices

The damping,C, and mass,M, matrices are generated for each finite element andassembled to the global matrices according to the topology in the same way as theelement stiffness matrices.

Since the analysis is performed in the time domain, viscous damping is applied cf.Section 5.3.4. Hence, the damping matrix is calculated by the use of (5.5)

Ce =βKe (C.2)

where the damping coefficientβ is given by (5.10) on page 31.

139

Appendix C. Enhancements to FE Winkler model

The soil springs and the gaskets are modelled as massless. Therefore, their elementmass matrices are 12x12 zero-matrices. For the tunnel beam elements, the consistentelement mass matrix,Me,cons, can be calculated from the shape functions, similar towhat has been done in Section B.2.2 with the help of (B.4). Theconsistent mass matrixends up to

Me,cons =µL

13

0 0 0 0 0 16

0 0 0 0 0

0 1335

0 0 0 11L210

0 970

0 0 0 −13L420

0 0 1335

0 −11L210

0 0 0 970

0 13L420

0

0 0 0 13

0 0 0 0 0 16

0 0

0 0 −11L210

0 L2

1050 0 0 −13L

4200 −L2

1400

0 11L210

0 0 0 L2

1050 13L

4200 0 0 −L2

14016

2 0 0 0 0 13

0 0 0 0 0

0 970

0 0 0 13L420

0 1335

0 0 0 −11L210

0 0 970

0 −13L420

0 0 0 1335

0 11L210

0

0 0 0 16

0 0 0 0 0 13

0 0

0 0 13L420

0 −L2

1400 0 0 11L

2100 L2

1050

0 −13L420

0 0 0 −L2

1400 −11L

2100 0 0 L2

105

The lumped mass matrix,Me,lmp, is calculated by the “row-sum” method, where thelumped mass matrix is formed by adding the off-diagonal entries in each row of theconsistent matrix to the diagonal entry, according to Zienkiewicz & Taylor (1989)

Me,lmp =µL diag

12

12+

3L140

12−

3L140

12

−L12

+L2

420L12

+L2

42012

12−

3L140

12+

3L140

12

L12

+L2

420

−L12

+L2

420

(C.3)

The lumped and consistent matrices are combined with (B.6) on page 137.

C.2 Time integration

The numerical integration of the equations of motion, (C.1), is performed with a New-mark family algorithm. This family of algorithms is widely used in structural dy-namics. To ensure unconditional stability, the Crank-Nicolson algorithm, which is aspecial case of the Newmark algorithm with the Newmark parameters(β,γ) = ( 1

4, 1

2),

has been employed. The time step is∆t = 0.01s as for the continuum model cf.Section 9.1.4. The Newmark family of algorithms has been documented by e.g.Nielsen (2007).

140

Forced displacements

C.3 Forced displacements

Since the earthquake does not influence the system through external loads, but throughforced displacements as described in Section 4.4, the forces in the load vectorf(t ) areunknown at the degrees of freedom with forced displacement.In stead, the forceddisplacements are introduced into the set of equations, thus rewriting (C.1) to

[

K11 K12

K21 K22

][

u1

u2

]

+

[

C11 C12

C21 C22

][

u1

˙u2

]

+

[

M11 M12

M21 M22

][

u1

ü2

]

=

[

f1

f2

]

(C.4)

where a bar( ) denotes a prescribed value,i.e. the known forced displacements, andthe known forces in the degrees of freedom without prescribed displacements. All ofthese forces is zero in the present case.

The equations of (C.4) with prescribed forces can now be rewritten to

K11u1 +C11u1 +M11u1 =(

f1 −K12u2 −C12˙u2 −M12

ü2

)

(C.5)

Ku+ C ˙u+M ü = f (C.6)

where tilde( ) denotes reduced vectors and matrices, the definition of which shouldbe clear from the rewriting of (C.5) to (C.6). The similaritybetween (C.6) and thebasic equation of motion (C.1) should be noted. This form is easily implemented intostandard Newmark schemes.

C.4 Spring elements

Two new elements are introduced in the program: the soil spring element and thegasket element. Both are built on the basis of the three-dimension beam element,simply by editing the generated element matrices. The element matrices of the beamelement are 12x12 matrices, since the beam element consistsof three translational andthree rotational degrees of freedom in each beam end. The stiffness matrix for thespring element is changed to a new 12x12 matrix, with the spring stiffnessk in entries1 and 7. Thus, only the axial translational degrees of freedom provide any stiffness.

Ke,spring =

k 0 · · · 0 −k

0 0 0...

......

0 0 0

−k 0 · · · 0 k

0(7x5)

0(5x7)

0(5x5)

(C.7)

Similarly, the gasket element is constructed from the beam element stiffness ma-trix. Since the shearing stiffness enters, uncoupled with other degrees of freedom,the 12x12 matrix is constructed with the longitudinal spring stiffnesskl in entries 1and 7, and the shear spring stiffnessks in entries 2, 3, 8 and 9. Thus, all translational

141

Appendix C. Enhancements to FE Winkler model

degrees of freedom in the two element ends are coupled.

Ke,spring =

kl

ks

ks

0(3x3)

−kl

−ks

−ks

0(3x3)

0(3x3)

0(3x3)

0(3x3)

0(3x3)

−kl

−ks

−ks

0(3x3)

kl

ks

ks

0(3x3)

0(3x3)

0(3x3)

0(3x3)

0(3x3)

(C.8)

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IX

DCROSS SECTION DATA

In this appendix, the data for the cross section of the immersed tunnel is outlined. Thedata is used throughout the thesis,i.a. in the Winkler model.

A typical cross section is presented in Figure 5.2 on page 28.For the present thesis,some simplifications of the geometry of the tunnel has been made. The cross sectionused in the thesis is presented in Figure D.1.

53006100

14800

1100600

1500

34500

8700

y

z

Figure D.1: Definition sketch of typical cross section of the tunnel. The dotted line around theperimeter shows the location of the Gina gasket. The section is both horizontaland vertical symmetric about the centre of gravity. All measures in mm. AfterCOWI (2007).

143

Appendix D. Cross section data

D.1 Length of gasket

Around the perimeter of the cross section, the Gina gasket islocated. The total lengthLgasket of the gasket is

Lgasket = 84.4m

D.2 Area

The full areaAfull and the solid areaAsolid of the cross section of the tunnel can becalculated according to Figure D.1 to

Afull = 300.2m2

Asolid = 111.6m2

D.3 Second moment of area

The second moments of area around the centre of gravity are calculated as

Iy y =1

12

(

34.5m · (8.7m)3−2 ·14.8m · (6.1m)3

−1.5m · (5.3m)3)

= 1315m4

Izz =1

12

(

8.7m · (34.5m)3−5.3m · (1.5m)3

)

−2(

112

·6.1m · (14.8m)3+ (8.75m)2

·6.1m ·14.8m)

= 12650m4

144

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IX

EROTATION MATRICES

To obtain results from the Winkler model, it is needed to rotate a vector in space, asit is described in Section 8.4.2. The rotation is obtained with transformation matrices.The notation follows Figure 8.13 on page 65.

The direction vector~P is transformed into the rotated direction vector~Prot by multi-plication with the transformation matricesRz , Ry andRx .

~Prot = Rz ·Ry ·Rx ·~P (E.1)

It should be noted that the order of the matrix multiplications in (E.1) in principle isnot trivial. In general, it matters, which rotation is applied first, but in the present casethe rotations are very small. In practical use, this makes the order of multiplicationunimportant.

The rotation matrices are bye.g.Goldstein (1980, pp146-147) given as:

Rz =

cos(θz ) −sin(θz ) 0

sin(θz ) cos(θz ) 0

0 0 1

(E.2)

Ry =

cos(θy ) 0 sin(θy )

0 1 0

−sin(θy ) 0 cos(θy )

(E.3)

Rx =

1 0 0

0 cos(θx ) −sin(θx )

0 sin(θx ) cos(θx )

(E.4)

145

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